On the measure and the structure of the free boundary of the lower dimensional obstacle problem

TL;DR

This paper thoroughly analyzes the structure and measure-theoretic properties of the free boundary in the lower dimensional obstacle problem, providing classifications and rectifiability results.

Contribution

It offers a detailed description of the free boundary, proving its local finiteness, rectifiability, and classifying blow-ups and frequencies for almost every free boundary point.

Findings

Finite $(n-1)$-dimensional Hausdorff measure of the free boundary

Rectifiability of the free boundary

Classification of blow-ups and frequencies

Abstract

We provide a thorough description of the free boundary for the lower dimensional obstacle problem in $\mathbb{R}^{n+1}$ up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, we prove (i) local finiteness of the $(n-1)$-dimensional Hausdorff measure of the free boundary, (ii) $\mathcal{H}^{n-1}$-rectifiability of the free boundary, (iii) classification of the frequencies up to a set of dimension at most (n-2) and classification of the blow-ups at $\mathcal{H}^{n-1}$ almost every free boundary point.

Equations

$$E^{+}:=E\cap\big{\{}x\in\mathbb{R}^{n+1}:\,x_{n+1}>0\big{\}}E^{\prime}:=E\cap\big{\{}x_{n+1}=0\big{\}}.$$

$$\displaystyle\mathscr{A}_{R}:=\Big{\{}v\in H^{1}(B_{R},|x_{n+1}|^{a}{\mathcal{L}}^{n+1}):\,v(x^{\prime},x_{n+1})=v(x^{\prime},-x_{n+1})v(x^{\prime},0)\geq 0\Big{\}}.$$

$$\begin{cases}u(x^{\prime},0)\geq 0&\text{for }\;(x^{\prime},0)\in B_{R}^{\prime},\\ u(x^{\prime},x_{n+1})=u(x^{\prime},-x_{n+1})&\text{for }\;x=(x^{\prime},x_{n+1})\in B_{R},\\ \textup{div}\big{(}|x_{n+1}|^{a}\nabla u(x)\big{)}=0&\text{for }\;x\in B_{R}\setminus\big{\{}(x^{\prime},0)\,:\,u(x^{\prime},0)=0\big{\}},\\ \textup{div}\big{(}|x_{n+1}|^{a}\nabla u(x)\big{)}\leq 0&\text{in the sense of distribution in }\,B_{R},\\ u(x)=g(x)&\text{for }\;x\in\partial B_{R},\end{cases}$$

$${\int }_{B_R}|\nabla v{|}^2|x_{n+1}{|}^a\mathrm{d}x$$

$$\Lambda(u):=\big{\{}(x^{\prime},0)\in B_{R}^{\prime}\,:u(x^{\prime},0)=0\big{\}},$$

$$s:=\frac{1-a}{2}$$

$$\displaystyle\bar{u}_{x_{0},r}(y):=\frac{r^{\frac{n+a}{2}}u(x_{0}+r\,y)}{\Big{(}\int_{\partial B_{r}}u^{2}\,|x_{n+1}|^{a}\,{\rm d}{\mathcal{H}}^{n}\Big{)}^{\nicefrac{{1}}{{2}}}}\quad\quad\forall\;y\in B_{\frac{R-|x_{0}|}{r}}.$$

$$\Gamma (u)=\text{Reg}(u)\cup \text{Sing}(u)\cup \text{Other}(u),$$

$${\mathcal{L}}^{n+1}\big{(}{\mathcal{T}}_{r}(\Gamma(u)\cap K)\big{)}\leq C(K)\,r^{2}\quad\forall\;r\in(0,1),$$

$${\mathcal{H}}^{n-1}\Big{(}\Gamma(u)\setminus\bigcup_{i\in{\mathbb{N}}}M_{i}\Big{)}=0.$$

$$\{2m,2m-1+s{\}}_{m\in \mathbb{N}\setminus \{0\}}\subseteq J\subseteq \{1+s\}\times [2,\infty ).$$

$$\lambda (x_0)\in \{2m,2m-1+s,2m+2s{\}}_{m\in \mathbb{N}\setminus \{0\}}\forall x_0\in \Gamma (u)\setminus \Sigma (u).$$

$$\textup{BU}(x_{0})=\Big{\{}\bar{h}_{\lambda(x_{0})}(x\cdot e_{x_{0}},x_{n+1})\Big{\}},$$

$${\mathcal{H}}^{n-1}\big{(}\Gamma(u)\cap K\big{)}<+\infty\quad\forall\;K\subset\subset\mathbb{R}^{n}.$$

$$\Vert {\nabla }_{\tau }u{\Vert }_{C^s(B_{\text{nicefrac}12})}+\Vert \text{sign}(x_{n+1})|x_{n+1}{|}^a{\partial }_{x_{n+1}}u{\Vert }_{C^{\alpha }(B_{\text{nicefrac}12})}\le C_{\text{ref}t:reg}\Vert u{\Vert }_{L^2(B_1,|x_{n+1}{|}^a{\mathcal{L}}^{n+1})},$$

$$\Vert u^{+}{\Vert }_{C^0(B_{\text{nicefrac}34})}+\Vert u^{-}{\Vert }_{C^0(B_{\text{nicefrac}34})}\le C\Vert u{\Vert }_{L^2(B_1,|x_{n+1}{|}^a{\mathcal{L}}^{n+1})},$$

$$\underset{x\in B_{\rho }}{\sup }|\nabla u(x)\cdot x|\le C{\rho }^{-\frac{n+1+a}{2}}\Vert u{\Vert }_{L^2(B_{2\rho },|x_{n+1}{|}^a{\mathcal{L}}^{n+1})},$$

$$\underset{x_{n+1}\downarrow 0^{+}}{\lim }{x}_{n+1}^a{\partial }_{n+1}u(x^{\prime },x_{n+1})=0$$

$$\underset{x_{n+1}\downarrow 0^{+}}{\lim }{x}_{n+1}^a{\partial }_{n+1}u(x^{\prime },x_{n+1})\le 0$$

$$u(x)\text{div}(|x_{n+1}{|}^a\nabla u(x))=0$$

$$T(\phi (x))$$

$$=2\underset{ϵ\downarrow 0}{\lim }{\int }_{\{x_{n+1}\ge ϵ\}\cap B_1}\nabla u(x)\cdot \nabla \phi (x)|x_{n+1}{|}^a\mathrm{d}x$$

$$\stackrel{\text{eqref}e:ob-pblocal}{=}-2\underset{ϵ\downarrow 0}{\lim }{\int }_{\{x_{n+1}=ϵ\}\cap B_1}{\partial }_{n+1}u(x^{\prime },ϵ)\phi (x^{\prime },ϵ){ϵ}^a\mathrm{d}{x}^{\prime }$$

$$=-2{\int }_{B_1^{\prime }}f(x^{\prime },0)\phi (x^{\prime },0)\mathrm{d}{x}^{\prime }.$$

$$\text{div}(|x_{n+1}{|}^a\nabla u(x))=-2f(x^{\prime },0){\mathcal{H}}^n\text{vrule}height=7.0pt,width=0.5pt,depth=0.0pt\text{vrule}height=0.5pt,width=6.0pt,depth=0.0pt{B}_1^{\prime }.$$

$$\varphi (t):=\{\begin{array}{ll}1& \text{for }0\le t\le \frac{1}{2},\\ 2(1-t)& \text{for }\frac{1}{2}<t\le 1,\\ 0& \text{for }1<t.\end{array}$$

$$I_u(x_0,r):=\frac{r{D}_u(x_0,r)}{H_u(x_0,r)}\forall r<R-|x_0|,$$

$$D_{u}(x_{0},r):=\int\phi\big{(}\textstyle{\frac{|x-x_{0}|}{r}}\big{)}\,|\nabla u(x)|^{2}\,|x_{n+1}|^{a}{\rm d}x,$$

$$H_{u}(x_{0},r):=-\int\phi^{\prime}\Big{(}\textstyle{\frac{|x-x_{0}|}{r}}\Big{)}\,\frac{u^{2}(x)}{|x-x_{0}|}\,|x_{n+1}|^{a}\,{\rm d}x.$$

$$E_{u}(x_{0},r):=\int-\phi^{\prime}\Big{(}\textstyle{\frac{|x-x_{0}|}{r}}\Big{)}\frac{|x-x_{0}|}{r^{2}}\Big{(}\nabla u(x)\cdot\frac{x-x_{0}}{|x-x_{0}|}\Big{)}^{2}\,|x_{n+1}|^{a}\,{\rm d}x.$$

$$v(y):=cu(x_0+ry)$$

$$I_{u}(x_{0},r_{1})-I_{u}(x_{0},r_{0})=\int_{r_{0}}^{r_{1}}\frac{2\,t}{H_{u}^{2}(x_{0},t)}\Big{(}H_{u}(x_{0},t)\;E_{u}(x_{0},t)-D_{u}^{2}(x_{0},t)\Big{)}\,{\rm d}t$$

$$\displaystyle D_{u}(x_{0},t)=-\frac{1}{t}\int\phi^{\prime}\big{(}\textstyle{\frac{|x-x_{0}|}{t}}\big{)}\,u(x)\nabla u(x)\cdot\frac{x-x_{0}}{|x-x_{0}|}\,|x_{n+1}|^{a}\,{\rm d}x,$$

Full text

• (i)

local finiteness of the $(n-1)$-dimensional Hausdorff measure of the free boundary,

• (ii)

${\mathcal{H}}^{n-1}$-rectifiability of the free boundary,

• (iii)

classification of the frequencies up to a set of dimension at most $(n-2)$ and classification of the blow-ups at ${\mathcal{H}}^{n-1}$ almost every free boundary point.

E. S. has been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) through a Visiting Professor Fellowship. E. S. is very grateful to the DiMaI “U. Dini” of the University of Firenze for the support during the visiting period. M. F. is a member of the GNAMPA of INdAM. He warmly thanks the University of Leipzig for the support during a visit when part of this work was conceived.

1. Introduction

Thin obstacle-type problems naturally appear in several models of applied sciences, such as contact mechanics (cf. the classical Signorini problem) and, as pointed out more recently, in free boundary problems for fractional diffusions, such as quasi-geostrophic flows, American options’ pricing, anomalous diffusions etc… Due to their character of prototypical nonlinear and non-local equations, in the recent years this class of problems has been intensively studied, culminating in several important contributions and breakthroughs (cf., e.g., [4, 11, 5, 12, 23, 14, 29, 7, 20, 3]). Nevertheless, many important questions are not yet answered, most importantly the ones concerning the global structure of the free boundary, which according to the available results in the literature is not excluded to have infinite measure or to be fractal, already in the simplest model cases.

Here we answer to this and to other related questions, such as the uniqueness of blow-ups and the structure of the free boundary for solutions to the thin obstacle problem, giving a complete description of the top-stratum of the free boundary up to a set of ${\mathcal{H}}^{n-1}$-measure zero. These results are new also in the framework of the classical Signorini problem in elasticity (for the antiplane case) and they are obtained by a combination of analytical and geometric measure theory arguments which can be suitably exploited also for similar free boundary type problems.

1.1. The problem

In this article we consider a class of lower dimensional obstacle problems. In order to state them, for any subset $E\subset\mathbb{R}^{n+1}$ we set

[TABLE]

For any point $x\in\mathbb{R}^{n+1}$ we will write $x=(x^{\prime},x_{n+1})\in\mathbb{R}^{n}\times\mathbb{R}$. Moreover, $B_{r}(x)\subset\mathbb{R}^{n+1}$ denotes the open ball centered at $x\in\mathbb{R}^{n+1}$ with radius $r>0$, and $\overline{B_{r}}(x)$ its closure (we omit to write the point $x$ if the origin). For every $R>0$, we denote by $\mathscr{A}_{R}$ the set of functions in the weighted Sobolev space $H^{1}(B_{R},|x_{n+1}|^{a}{\mathcal{L}}^{n+1})$, with $a\in(-1,1)$, which are even symmetric with respect to $x_{n+1}$ and which have positive traces on $B_{R}^{\prime}$:

[TABLE]

The thin obstacle problems we consider are then the following:

[TABLE]

where $g\in\mathscr{A}_{R}$ is a given boundary value datum. Note that (1.1) are the Euler–Lagrange equations satisfied by the unique minimizer of the energy

[TABLE]

in the class \mathscr{A}_{R,g}:=\mathscr{A}_{R}\cap\big{\{}g+H^{1}_{0}(B_{R},|x_{n+1}|^{a}{\mathcal{L}}^{n+1})\big{\}}. In particular, in case $a=0$ problem (1.1) corresponds to the well-known scalar Signorini problem. We denote by $\Lambda(u)$ the coincidence set of a solution $u$,

[TABLE]

and by $\Gamma(u)$ its free boundary, which is the topological boundary of $\Lambda(u)$ in the relative topology of $B_{R}^{\prime}$. In order to avoid unnecessary complications, in this work we consider the case of zero obstacle prescribed on flat hypersurfaces only. Nevertheless, the techniques developed in the paper can be generalized to consider non-constant and non-flat obstacles, as well as for other free boundary problems (such as the fractional obstacle problem, for which the analogous results of this paper are going to appear in a future work). Moreover, we set

[TABLE]

throughout the whole paper.

1.2. A short survey of the existing literature

In the last years there has been an intensive research activity in trying to set up the regularity properties of the solutions to (1.1) and the corresponding free boundaries. We resume in what follows the state of the art for what concerns the zero obstacle case. To this aim we introduce the following notation for the rescalings of a solution $u$: for every $x_{0}\in\Gamma(u)$ and $r>0$, we set

[TABLE]

By [12, Section 6] the collection of functions $\{\bar{u}_{x_{0},r}\}_{r>0}$ is pre-compact in the weighted Sobolev space $H^{1}_{\textup{loc}}(\mathbb{R}^{n+1},|x_{n+1}|^{a}\,{\mathcal{L}}^{n+1})$. Their limiting points are called blow-ups of $u$ at $x_{0}$ and are homogeneous functions, whose homogeneity depends only on $x_{0}$ and not on the extracted subsequence (for a proof see also Corollary 2.10 and Remark 2.14 below). The set of all blow-ups of a solution $u$ at $x_{0}$ is denoted by $\textup{BU}(x_{0})$, and their common homogeneity $\lambda(x_{0})$ is called the infinitesimal homogeneity or the frequency of $u$ at $x_{0}$ (this is indeed the limiting value, as the radius vanishes, of an Almgren’s type frequency function).

The following statements summarize several results available in the current literature.

A. Optimal regularity of $u$. The solutions $u$ to (1.1) are one-sided $C^{1,s}$, $s=\nicefrac{{(1-a)}}{{2}}$. More precisely, $u\in\textrm{Lip}(B_{1})\cap C^{1,s}(B_{1}^{\pm}\cup B_{1}^{\prime})$, as proved by Athanasopoulos and Caffarelli [4] for $a=0$, and by Caffarelli, Salsa and Silvestre [12] for all $a\in(-1,1)$ (see also [21, 22, 9, 28, 36, 37, 35] for previous results).

B. Free boundary regularity. The free boundary $\Gamma(u)$ can be split as:

[TABLE]

with these subsets being pairwise disjoint, and more precisely

• (i)

$\textup{Reg}(u)$ is the subset of points in $\Gamma(u)$ in which blow-ups are $(1+s)$-homogeneous. $\textup{Reg}(u)$ is relatively open in $\Gamma(u)$ and it is an analytic $(n-1)$-dimensional submanifold of $\mathbb{R}^{n+1}$ (the $C^{1,\alpha}$ regularity has been shown in [5, 12] – see also [20, 25] for a different proof based on the epiperimetric inequality; higher regularity follows from [14, 29]);

• (ii)

$\textup{Sing}(u)$ is the subset of points in $\Gamma(u)$ for which the blow-ups are $2m$-homogeneous. In the case of the Signorini problem $a=0$, they are also characterized by the fact that their contact sets have density zero with respect to ${\mathcal{H}}^{n}$. Furthermore, in such a case Garofalo and Petrosyan [23] proved that $\textup{Sing}(u)$ is contained in a countable union of $C^{1}$-regular $(n-1)$-dimensional submanifolds.

C. Blow-up analysis. The blow-ups of $u$ at a free boundary point $x_{0}$ satisfy the ensuing properties:

• (i)

$\textup{BU}(x_{0})\subseteq{\mathcal{H}}_{\lambda(x_{0})}$, the latter set being the positive cone of $\lambda(x_{0})$-homogeneous local solutions to (1.1) even with respect to $x_{n+1}$. Moreover, the possible values of the frequency $\lambda(x_{0})$ lie in the set $\{1+s\}\times[2,+\infty)$ (cf. [12]);

• (ii)

the blow-ups are unique both at every point of $\textup{Reg}(u)$ (cf. [12]), and at every point of $\textup{Sing}(u)$ for the Signorini problem $a=0$ (cf. [23]).

Despite these significant achievements, many issues on the analysis of the regularity of the free boundary and the corresponding blow-ups of solutions to (1.1) remain still unsolved, even for the scalar Signorini problem. The most striking fact is that nothing is known about the global nature of the free boundary, which in principle is not known to have the right dimensionality of a boundary in $\mathbb{R}^{n}\times\{0\}$ (i.e., $n-1$), nor it is known to retain any boundary-like structure (as far as we know, $\Gamma(u)$ can be even fractal). In particular, there are no results about the subset of free boundary points $\textup{Other}(u)$, which are neither regular nor singular (according to the definitions in literature). On the other hand, explicit examples show that $\textup{Other}(u)$ is in general not empty, and indeed it may coincide with the full free boundary (cf. § 8)!

1.3. The main results of the paper

In this paper we answer to some of the questions mentioned above, such as that concerning the dimension of the free boundary, and we give a comprehensive description of the set $\textup{Other}(u)$ and $\textup{Sing}(u)$ in the general case $a\in(-1,1)$ up to a null ${\mathcal{H}}^{n-1}$ set. Our results are already new for the case of the Signorini problem $a=0$ and extend in various directions what was previously known. In particular, the short outcome of our analysis is the global picture of the free boundary of the thin obstacle problem as an $(n-1)$-dimensional set with locally finite measure (in fact with finite Minkowski content) satisfying almost everywhere a similar stratification as for the classical obstacle problem (including some uniqueness results of the blow-ups), cf. [8, 10, 38, 30].

We start off showing that the free boundary is $(n-1)$-dimensional in a strong measure theoretic sense.

Theorem 1.1**.**

Let $u$ be a solution to the thin obstacle problem (1.1) in $B_{1}$. Then, the free boundary $\Gamma(u)$ has locally finite $(n-1)$-dimensional Minkowski content: i.e., for every $K\subset\subset B_{1}^{\prime}$ there exists a constant $C(K)>0$ such that

[TABLE]

where ${\mathcal{T}}_{r}(E):=\{x\in\mathbb{R}^{n+1}:{\textup{dist}}(x,E)<r\}$ for all $E\subseteq\mathbb{R}^{n}$.

Next, we prove the following geometric regularity result for the free boundary establishing its ${\mathcal{H}}^{n-1}$-rectifiability.

Theorem 1.2**.**

Let $u$ be a solution to the thin obstacle problem (1.1) in $B_{1}$. Then, there exist at most countably many $C^{1}$-regular submanifolds $M_{i}$ of dimension $n-1$ in $\mathbb{R}^{n+1}$ such that

[TABLE]

The last result concerns one of the major open question in the field, namely to determine the possible values of the frequency $\lambda(x_{0})$, or equivalently the smallest set $J\subset(0,\infty)$ for which $\textup{BU}(x_{0})\subseteq\cup_{\lambda\in J}{\mathcal{H}}_{\lambda}$ for every $x_{0}\in\Gamma(u)$, recall that ${\mathcal{H}}_{\lambda}$ denotes the set of $\lambda$-homogeneous solutions to (1.1). As explained in C. (i) above, it is known that

[TABLE]

Moreover, by definition $\lambda(x_{0})=1+s$ for all $x_{0}\in\textup{Reg}(u)$ and $\lambda(x_{0})\in 2{\mathbb{N}}\setminus\{0\}$ for all $x_{0}\in\textup{Sing}(u)$. In the following theorem we make a step forward to clarify this stage.

Theorem 1.3**.**

Let $u$ be a solution to the lower dimensional obstacle problem (1.1) in $B_{1}$. Then, there exists a subset $\Sigma(u)\subset\Gamma(u)$ with Hausdorff dimension at most $n-2$ such that

[TABLE]

In addition, for ${\mathcal{H}}^{n-1}$-a.e. point $x_{0}\in\Gamma(u)\setminus\Sigma(u)$ with frequency $\lambda(x_{0})\in\{2m,2m-1+s\}_{m\in{\mathbb{N}}\setminus\{0\}}$ the blow-up of $u$ at $x_{0}$ is unique and depends on two variables only: namely,

[TABLE]

for some $e_{x_{0}}\in\mathbb{R}^{n+1}$ with $|e_{x_{0}}|=1$ and $e_{x_{0}}\cdot e_{n+1}=0$, and $\bar{h}_{\lambda(x_{0})}$ uniquely determined by $\lambda(x_{0})$.

1.4. Comments on the main results

A few remarks are in order.

1.4.1. Finite measure

The main consequence of Theorem 1.1 is that the free boundary has locally finite ${\mathcal{H}}^{n-1}$ measure:

[TABLE]

Nevertheless, the estimate on the Minkowski content is significantly stronger: among the other consequences, (1.4) implies, for instance, that the free boundary is nowhere dense. In addition, Theorem 1.2 establishes that the free boundary is a ${\mathcal{H}}^{n-1}$-rectifiable set, a piece of information which cannot be deduced nor implies the estimate on the Hausdorff measure (1.6).

The estimate on the Hausdorff dimension of the free boundary can be deduced independently from Theorem 1.1 and Theorem 1.2 by a different and more direct stratification argument (cf. Theorem 8.1).

1.4.2. Structure of the free boundary

Theorem 1.2 extends the analysis of the structure of the free boundary points to a subset of full measure of $\textup{Sing}(u)\cup\textup{Other}(u)$. Note that the structure of the points in $\textup{Sing}(u)$ for $a\neq 0$ had not been dealt with before in the literature. Nevertheless, Theorem 1.2 does not imply the pointwise results in B. (i) & (ii) for $\textup{Reg}(u)$ and $\textup{Sing}(u)$, for the latter set if $a=0$, because we prove a measure theoretic regularity property for $\Gamma(u)$, namely its ${\mathcal{H}}^{n-1}$-rectifiability (cf. (1.5)).

1.4.3. Frequency

Points with frequencies $2m-1+s$ and $2m+2s$, with $m\in{\mathbb{N}}\setminus\{0,1\}$, belong to $\textup{Other}(u)$, though it is not known whether they do exhaust such a set or not in general. In other words, the problem of classifying all possible frequencies for free boundary points is settled by Theorem 1.3 only up to sets of dimension at most $n-2$, but it remains open pointwise.

Moreover, if on one hand there are examples of free boundary points with frequency $2m$ and $2m-1+s$, on the other hand there are no examples of points with frequency $2m+2s$. In dimension $n=1$ one can show that such points do not exist, that is $\textup{Other}(u)=\{2m-1+s\}_{m\in{\mathbb{N}}\setminus\{0,1\}}$ and also that $\Sigma(u)=\emptyset$ (see § 8 – the case $a=0$ has been discussed in [23]). In higher dimensions it is then natural to conjecture the same results.

The reason why we are unable to rule out points with frequencies $2m+2s$ if $n\geq 2$ is related to the existence of $(2m+2s)$-homogeneous solutions with contact set $\Lambda=\mathbb{R}^{n}\times\{0\}$, which potentially could arise as blow-ups in a free boundary point (with the free boundary disappearing in the limit). This possibility might seem an apparent and striking discrepancy with the measure estimate and the structure result of Theorems 1.1 and 1.2 and make these results in some sense surprising (see § 8.3 for further comments).

Finally, the estimate on the Hausdorff dimension of $\Sigma(u)$ follows from its inclusion in the subset of points of $\Gamma(u)$ whose blow-ups have at most $(n-2)$ directions of invariance, for such a set the dimensional estimate is actually sharp (cf. Theorem 8.1).

1.4.4. Blow-ups

The uniqueness of blow-ups provided by Theorem 1.3 at points of the free boundary with frequency $2m$ and $2m-1+s$ univocally describes the infinitesimal behaviour of the solution $u$. In particular, it shows that the solutions look locally like a homogeneous function of a single horizontal variable and of $x_{n+1}$. Note that, for any different choice of the renormalization of the rescalings (1.2), either the limit does not exist or it is a multiple of $\bar{h}_{\lambda}$, thus justifying the notion of unique limiting profile.

For the prospective points with frequency $2m+2s$, as a by-product of the results in Appendix A, we are also able to classify all possible blow-ups.

1.5. Concerning the proofs

Our analysis is based on geometric measure theory techniques, which exploit and develop some ideas recently introduced in the context of minimal surfaces theory. The point of view we adopt is new in the theory of free boundaries and we believe it has many potentialities for other related problems.

The proof is based on the ideas and the techniques recently introduced by Naber–Valtorta [31, 32] in the context of minimal surfaces and harmonic maps. The main ingredients of our study are (a variant of) Almgren’s frequency function and the Peter Jones’ number $\beta^{(n-1)}_{\mu}$ pertaining to a suitable measure $\mu$ supported on the free boundary $\Gamma(u)$ (the terminology mean flatness is also adopted in literature to term the $\beta$-numbers, since they provide an integral control of the flatness of the support of the underlying measure $\mu$, see [2]). The starting point is the striking observation by Naber–Valtorta [31, 32] that the square power of the mean flatness can be controlled by an average of the oscillations of a monotone density. Indeed, when this happens, a careful covering argument [31, 32], and recently developed rectifiability criteria by David–Toro [13], Azzam–Tolsa [6] and Naber–Valtorta [31, 32] lead to the local finiteness and the rectifiability of the singular sets of minimal surfaces and harmonic maps (see also the paper by De Lellis, Marchese, Spadaro and Valtorta [16] for an extension to a special case in higher co-dimension).

For our analysis of the thin obstacle problem, we generalize and develop these approaches. The starting point is an estimate of the mean flatness with respect to a Borel measures $\mu$ supported on the free boundary with the spatial oscillation of Almgren’s frequency function (cf. Proposition 4.2). Note that the case of the frequency function is different from the mass ratio of a minimal surface, because the renormalization factor is intrinsically defined by the solution itself (usually a variant of the $L^{2}$-norm at the boundary of a ball), instead of being purely dimensional. This requires a novel estimate for the frequency of the solutions to the lower dimensional obstacle problem, which is based on a different set of spatial variations and is proven in Proposition 3.3: here we follow closely ideas of [16], where an analogous estimate is proved in the context of multiple-valued functions as a result of this spatial variations of the frequency.

A careful analysis of the rigidity properties of homogeneous solutions to the thin obstacle problem (1.1) (cp. Proposition 5.6) is then necessary for our argument. To this aim, as in general no growth estimate from below for solutions from the free boundary are at disposal, it is mandatory for us to introduce the set of nodal points.

Using such rigidity results and the mentioned estimate on the mean flatness via the frequency we use the covering argument and the discrete Reifenberg theorem by Naber–Valtorta in [31, 32] in order to infer Theorem 1.1. Then, Theorem 1.2 is obtained by means of the rectifiability criterion recently established by Azzam–Tolsa [6] and indipendently by Naber–Valtorta in [31, 32], while Theorem 1.3 is a consequence of Almgren’s stratification principle (see, e.g., [19]) and the classification of homogeneous solutions of the PDE (1.1) given in § 8 and § A.

1.6. Structure of the paper

We start off introducing several preliminaries in § 2. More precisely, in § 2.1 we collect the results concerning the regularity of the solutions to the thin obstacle problem. In § 2.2 we introduce the variant of the frequency function we are going to use and we derive several useful properties. We then show in § 3 how to deduce from these an oscillation estimate of the frequency. The aforementioned control of the flatness of the free boundary (defined in terms of the Peter Jones’ numbers), with the oscillation of the frequency is established in Proposition 4.2. Next, § 5 is devoted to the classification results for homogeneous solutions to the PDE in (1.1) under several conditions and to study the rigidity properties of almost homogeneous solutions. Full proofs of the classification are provided in the Appendix A.

We then proceed with the proofs of Theorems 1.1, 1.2 and 1.3 in § 6, § 7 and § 8, respectively. In the corresponding section we recall the analytical results that we exploit in the proofs, namely the discrete Reifenberg theorem by Naber–Valtorta [31, 32], the rectifiability criterion by Azzam–Tolsa [6], and Almgren’s stratification principle following the abstract version provided in our paper in collaboration with Marchese [19].

2. Preliminaries on the thin obstacle problem

In this section we recall some of the known results on the thin obstacle problem.

2.1. Optimal regularity

The following is the main existence and regularity theorem by Caffarelli, Salsa and Silvestre [12].

Theorem 2.1**.**

For every $g\in\mathscr{A}_{1}$, there exists a unique solution $u$ to the thin obstacle problem (1.1) in $B_{1}$. Moreover, $\partial_{x_{i}}u\in C^{s}(B_{1})$ for $i=1,\ldots,n$, $|x_{n+1}|^{a}\partial_{x_{n+1}}u\in C^{\alpha}(B_{1})$, $0<\alpha<1-s$, and there exists a constant $C_{\ref{t:reg}}>0$ such that

[TABLE]

where $\nabla_{\tau}u=(\partial_{x_{1}}u,\ldots,\partial_{x_{n}}u)$ is the horizontal gradient.

Remark 2.2*.*

The estimates in [12, Proposition 4.3] are given in terms of the $C^{0}$ norm of $u$ on the right hand side of the inequality. Nevertheless, $u^{+}:=\max\{u,0\}$ and $u^{-}:=\max\{-u,0\}$ satisfy $\textup{div}(|x_{n+1}|^{a}\nabla u^{\pm}(x))\geq 0$ in $\mathcal{D}^{\prime}(B_{1}^{\prime})$. Therefore, by the $L^{\infty}$-estimate in [17, Theorem 2.3.1] we have that

[TABLE]

and then (2.1) follows by combining [12, Proposition 4.3] and (2.2).

Remark 2.3*.*

For later purposes we also need the estimate

[TABLE]

which follows straightforwardly from (2.1).

In particular, the function $u$ is analytic in \big{\{}x_{n+1}>0\big{\}}\cap B_{1} (see, e.g., [26]) and the following boundary conditions holds.

Corollary 2.4**.**

Let $u$ be a solution to the thin obstacle problem (1.1) in $B_{1}$. Then,

[TABLE]

Proof.

Set for simplicity $f(x^{\prime},0):=\lim_{x_{n+1}\downarrow 0^{+}}x_{n+1}^{a}\partial_{n+1}u(x^{\prime},x_{n+1})$, and note that by Theorem 2.1 we have that $f\in C^{\alpha}(B_{1}^{\prime})$, $0<\alpha<1-s$. By the even symmetry of $u$, for every $\varphi\in C^{1}_{c}(B_{1})$ even symmetric we get the following: let $T:=-\textup{div}(|x_{n+1}|^{a}\nabla u(x))$ in $\mathscr{D}^{\prime}(B_{1})$, then

[TABLE]

This shows that

[TABLE]

Thus, (2.4) and (2.5) follow directly from (1.1). Moreover, $u(x)\,\textup{div}(|x_{n+1}|^{a}\nabla u(x))$ is well-defined as a measure, and using (1.1) we also infer (2.6), because $f(x^{\prime},0)\,u(x^{\prime},0)=0$ for all $(x^{\prime},0)\in B_{1}^{\prime}$. ∎

2.2. The frequency function

As firstly noticed by Athanasopoulos, Caffarelli and Salsa in [5], one of the main quantities which are relevant to the analysis of the solutions to the thin obstacle problem is Almgren’s frequency function. Several variants of the frequency function have been introduced in the literature. For our purposes, we use the analog of that introduced in [15] in the context of higher co-dimension minimal surfaces.

Let $\phi:[0,+\infty)\to[0,+\infty)$ be the function given by

[TABLE]

We define the frequency of a solution $u$ to (1.1) at a point $x_{0}\in B_{R}^{\prime}$ by

[TABLE]

where

[TABLE]

and

[TABLE]

Note that the frequency is well-defined as long as $H_{u}(x_{0},r)>0$. As $H_{u}(x_{0},r)=0$ implies $u\equiv 0$ by the analyticity of $u$ in $B_{R}\setminus B_{R}^{\prime}$, we infer then that the frequency is always well-defined for non-trivial solutions $u$. For later convenience, we introduce also the notation

[TABLE]

In what follows, when $x_{0}=0$ we shall omit to write the base point $x_{0}$ in the notation of $I_{u}$, $D_{u}$, $H_{u}$ an $E_{u}$.

Remark 2.5*.*

The principal advantage of the frequency function $I_{u}(x_{0},r)$ is that it retains some average information of the solution $u$ on the annulus $B_{r}\setminus B_{\nicefrac{{r}}{{2}}}(x_{0})$, whereas the classical Almgren’s frequency function only involves the $L^{2}$ norm of $u$ on the sphere $\partial B_{r}(x_{0})$.

Remark 2.6*.*

If $u$ is a solution to the thin obstacle problem in $B_{R}$, then for every $r\in(0,R-|x_{0}|)$, $x_{0}\in B_{R}^{\prime}$ and for every $c>0$, the function $v:B_{1}\to\mathbb{R}$

[TABLE]

solves (1.1) in $B_{1}$ with respect to its own boundary conditions. Moreover, $I_{v}(0,\rho)=I_{u}(x_{0},\rho\,r)$ for every $\rho\in(0,1)$. This shows that the frequency function is scaling invariant, and in the sequel we will use this property repeatedly.

2.3. Monotonicity of the frequency

The following is a simple variant of the well-known monotonicity of the frequency (cf. [11]).

Proposition 2.7**.**

Let $u$ be a solution to the thin obstacle problem (1.1) in $B_{R}$. Then, for all $x_{0}\in\Gamma(u)\cap B_{R}^{\prime}$, the map $(0,R-|x_{0}|)\ni r\mapsto I_{u}(x_{0},r)$ is nondecreasing and

[TABLE]

for $0<r_{0}<r_{1}<R-|x_{0}|$. Moreover, $I_{u}(x_{0},\cdot)=\kappa$ for every $t\in(r_{0},r_{1})$ if and only if $u$ is $\kappa$-homogeneous with respect to $x_{0}$.

Proof.

We start off collecting some useful identities:

[TABLE]

To show (2.8), (2.9) and (2.10), we assume without loss of generality that $x_{0}=0$. For (2.8) we consider the vector field V(x):=\phi\big{(}\frac{|x|}{t}\big{)}\,u(x)\,\nabla u(x)\,|x_{n+1}|^{a}. Clearly $V$ has compact support, and $V\in C^{\infty}(\mathbb{R}^{n+1}\setminus B_{1}^{\prime},\mathbb{R}^{n})$ by Theorem 2.1. Moreover, for $x_{n+1}\neq 0$

[TABLE]

thus, $\lim_{y\downarrow(x^{\prime},0)}V(y)\cdot e_{n+1}=0$. Indeed, if $(x^{\prime},0)\in\Lambda_{\varphi}(u)$ it suffices to take into account the one-sided $C^{1,\alpha}$ regularity of $u$ in Theorem 2.1 to conclude

[TABLE]

Instead, if $(x^{\prime},0)\notin\Lambda_{\varphi}(u)$ we use (2.4) in Corollary 2.4. Thus, the distributional divergence of $V$ is the $L^{1}$ function given by

[TABLE]

Therefore, (2.8) follows from the divergence theorem by taking into account that $V$ is compactly supported.

Next (2.9) is a consequence of (2.8) and the direct computation

[TABLE]

Finally, to prove (2.10) we consider the vector field

[TABLE]

By Theorem 2.1 we have that $W\in C^{0}_{c}(B_{1},\mathbb{R}^{n})\cap C^{\infty}(B_{1}\setminus B_{1}^{\prime},\mathbb{R}^{n})$. Moreover, Corollary 2.4 implies that $W(x^{\prime},0)\cdot e_{n+1}=0$ for all $(x^{\prime},0)\in B_{1}^{\prime}$. Thus $\textup{div}\,W$ has no singular part in $B_{1}^{\prime}$, and we can compute pointwise

[TABLE]

Therefore, we infer that

[TABLE]

and we conclude (2.10) by direct differentiation

[TABLE]

By collecting (2.9) and (2.10), we finally compute the derivative of $\log I_{u}(t)$:

[TABLE]

In particular, identity (2.7) follows at once by multiplying by $I_{u}(t)$ and by integrating over $(r_{0},r_{1})$. In addition, by the Cauchy–Schwarz inequality, $r\mapsto I_{u}(r)$ is non-decreasing. Finally, if $I_{u}(t)=k$ for every $t\in(r_{0},r_{1})$, then

[TABLE]

In particular, by the equality case in the Cauchy–Schwarz inequality, we deduce that there exists a constant $\lambda\in\mathbb{R}$ such that

[TABLE]

i.e. $u(x)=|x|^{\lambda}u(\nicefrac{{x}}{{|x|}})$ for all $x\in B_{r_{1}}\setminus B_{\nicefrac{{r_{0}}}{{2}}}$. It then follows that $\lambda=k$ and by analyticity we conclude that $u$ is $k$-homogeneous in the whole $B_{R}$. ∎

From the monotonicity of the frequency, we infer the following consequences.

Corollary 2.8**.**

Let $u$ be a solution to the thin obstacle problem (1.1) in $B_{R}$. Then, for all $x_{0}\in\Gamma(u)\cap B_{R}^{\prime}$ and $0<r_{0}<r_{1}<R-|x_{0}|$, we have

[TABLE]

In particular, if $A_{1}\leq I(x_{0},t)\leq A_{2}$ for every $t\in(r_{0},r_{1})$, then

[TABLE]

Moreover,

[TABLE]

Proof.

The proof of (2.11) (and hence of (2.12) and (2.13)) follows from the differential equation (2.9). The proof of (2.14) is now a direct consequence:

[TABLE]

where in the last inequality we used that $H_{u}(x_{0},s)\leq H_{u}(x_{0},r)$ for $s\leq r$ by (2.13). ∎

2.4. Lower bound on the frequency and compactness

We first show that the frequency of a solution to (1.1) at free boundary points is bounded from below by a universal constant.

Lemma 2.9**.**

There exists a dimensional constant $C_{\ref{l:freq lower}}>0$ such that, for every solution $u$ to the thin obstacle problem (1.1) in $B_{R}$ and for every $x_{0}\in\Gamma(u)$, we have

[TABLE]

Proof.

By the co-area formula for Lipschitz functions we check that

[TABLE]

and

[TABLE]

An integration by parts then gives

[TABLE]

Therefore, we can conclude the lower bound (2.15) by using the Poincaré inequality in [12, Lemma 2.13]

[TABLE]

We can then give the following compactness result which will be instrumental for the analysis we develop. To this aim it is mandatory to introduce the nodal set of $u$:

[TABLE]

Notice that $\Gamma(u)\subseteq\mathcal{N}(u)$ by Corollary 2.4.

Corollary 2.10**.**

Let $(u_{k})_{k\in{\mathbb{N}}}$ be a sequence of solutions to the thin obstacle problem (1.1) in $B_{1}$, with $\sup_{k}I_{u_{k}}(1)<+\infty$, $H_{u_{k}}(1)\leq 1$ and $0\in\Gamma(u_{k})$ for every $k\in{\mathbb{N}}$. Then, there exist a subsequence $(u_{k_{j}})_{j\in{\mathbb{N}}}\subset(u_{k})_{k\in{\mathbb{N}}}$ and a solution $u_{0}$ to the thin obstacle problem in $B_{1}$ such that as $j\to\infty$

[TABLE]

Moreover, if there is a sequence of points $x_{k_{j}}\in\Gamma(u_{k_{j}})$ such that $x_{k_{j}}\to x_{0}\in B_{1}$, then

[TABLE]

Proof.

For every $t<1$, we have that

[TABLE]

where we have set for convenience $M:=\sup_{k}I_{u_{k}}(0,1)$. Moreover, from (2.14) we have that $\|u_{k}\|_{L^{2}(B_{1},|x_{n+1}|^{a}\,{\mathcal{L}}^{n+1})}\leq H_{u_{k}}(1)\leq 1$. The sequence $(u_{k})_{k\in{\mathbb{N}}}$ is equi-bounded in $H^{1}(B_{t},|x_{n+1}|^{a}\,{\mathcal{L}}^{n+1})$ for every $t<1$. Therefore, (2.18) – (2.21) follow from Theorem 2.1 (cf. also [12, Lemma 4.4]). Moreover, since $x_{k_{j}}\in\Gamma(u_{k_{j}})\subseteq\mathcal{N}(u_{k_{j}})$, (2.22) follows from (2.19)–(2.21). ∎

2.5. Blow-up profiles

An important consequence of the monotonicity of the frequency in Proposition 2.7 is the existence of blow-up profiles. For $u:B_{R}\to\mathbb{R}$ solution of (1.1) we introduce the rescalings

[TABLE]

Proposition 2.11**.**

Let $u$ be a solution to the thin obstacle problem (1.1) in $B_{R}$. Then, for every $x_{0}\in\Gamma(u)$ and for every sequence of numbers $(r_{j})_{j\in{\mathbb{N}}}\subset(0,1-|x_{0}|)$ with $r_{j}\downarrow 0$, there exists a subsequence $(r_{j_{k}})_{k\in{\mathbb{N}}}$ and function $u_{0}\in H^{1}_{\textup{loc}}(\mathbb{R}^{n+1},|x_{n+1}|^{a}{\mathcal{L}}^{n+1})$ such that $u_{0}$ satisfies (1.1), $u_{0}$ is homogeneous of degree $I(x_{0},0^{+})$ and

[TABLE]

Proof.

For every $\ell>0$, by Remark 2.6 we have $I_{u_{x_{0},r_{j}}}(\ell)=I_{u}(x_{0},r_{j}\,\ell)\leq I_{u}(x_{0},1-|x_{0}|)$. Therefore, from Corollary 2.8 we infer that there exists a constant $C=C(\ell)>0$ such that

[TABLE]

We can then use Corollary 2.10 and a diagonal argument to infer the existence of a subsequence $(r_{j_{k}})_{k\in{\mathbb{N}}}$ and a solution $u_{0}$ such that (2.24) holds. We only need to show that $u_{0}$ is homogeneous. To this aim we notice that, by taking into account Lemma 2.9, we have for every $\ell>0$

[TABLE]

In particular, by Proposition 2.7 we conclude the homogeneity of $u_{0}$ of degree $I_{u}(x_{0},0^{+})$. ∎

Corollary 2.12**.**

Let $u$ be a solution to the thin obstacle problem (1.1) in $B_{R}$ . Then,

[TABLE]

Proof.

We consider the rescaling $u_{x_{0},r_{j}}$ and a blow-up limit $u_{0}$. By Proposition 2.11 we know that $u_{0}$ is homogeneous of degree $I_{u}(x_{0},0^{+})$. Since solutions to (1.1) are $C^{1,s}(B_{R})$ (cf. [12]), we easily conclude that $I_{u}(x_{0},0^{+})\geq 1+s$ and (2.25) follows by monotonicity. ∎

Remark 2.13*.*

In general the limiting profile $u_{0}$ is not known to be unique. Uniqueness for ${\mathcal{H}}^{n-1}$-almost every free boundary point with infinitesimal homogeneity $2m$ and $2m-1+s$ will be established in Theorem 1.3, while uniqueness at every regular point follows from [12] (see also [20, 25] for an approach via the epiperimetric inequality) and at every singular point for $s=\nicefrac{{1}}{{2}}$ from [23].

Remark 2.14*.*

It is more common in the literature to define the blow-up rescalings $\bar{u}_{x_{0},r}$ as in (1.2). Nevertheless, by the same computations above, one can show that the height function $h_{u}(x_{0},t):=\int_{\partial B_{t}(x_{0})}u^{2}\,{\rm d}{\mathcal{H}}^{n}$ satisfies the analogous monotonicity properties of Corollary 2.8 (see [12]) and moreover by (2.16) it is comparable to $H_{u}(x_{0},t)$ (with a constant depending only on an upper bound of the frequency). In particular, this implies that the blow-ups with respect to these two different renormalizations only differ by a constant and all the results concerning them (e.g. the uniqueness) can be indifferently proven for either of the two definitions.

Due to our definition of the frequency, in the sequel we will always consider the rescalings defined in (2.23).

3. Main estimates on the frequency

In this section we prove the principal estimates on the frequency that we are going to exploit in the sequel.

Lemma 3.1**.**

For every $A>0$ there exists $C_{\ref{l:lim uniforme}}=C_{\ref{l:lim uniforme}}(A)>0$ such that, if $u$ is a solution to the thin obstacle problem (1.1) in $B_{2r}(x_{0})$, with $r>0$, $x_{0}\in\Gamma(u)$ and $I_{u}(x_{0},2r)\leq A$, then for every $x\in B^{\prime}_{\nicefrac{{r}}{{2}}}(x_{0})$

[TABLE]

Proof.

By rescaling it is enough to consider the case $x_{0}=0$, $r=1$ and $H_{u}(0,1)=1$ (cf. Remark 2.6). In oder to prove (3.1), we argue by contradiction: assume there exists functions $u_{k}$ and points $x_{k}\in B^{\prime}_{\nicefrac{{1}}{{2}}}$ contradicting the first inequality (3.1), i.e.

[TABLE]

Note that, since $I_{u_{k}}(0,2)\leq A$, it follows from (2.12) that $H_{u_{k}}(0,2)\leq 2^{n+a+2A}$. In particular, we can apply Corollary 2.10 and (up to passing to a subsequence, not relabeled), there exist $u_{\infty}$ and $x_{\infty}\in\bar{B}_{\nicefrac{{1}}{{2}}}$ such that $u_{k}\to u_{\infty}$ in $H^{1}_{\textup{loc}}(B_{2},|x_{n+1}|^{a}{\mathcal{L}}^{n+1})$ and $x_{k}\to x_{\infty}\in\bar{B}^{\prime}_{\nicefrac{{1}}{{2}}}$, with $u_{\infty}$ solution to the thin obstacle problem in $B_{R}$ for every $R<2$. By the strong convergence of $u_{k}$ to $u_{\infty}$ we then deduce that $H_{u_{\infty}}(x_{\infty},1)\in\{0,\infty\}\cap\mathbb{R}=\{0\}$. Given that $u_{\infty}$ is analytical in $B_{2}\setminus\{x_{n+1}=0\}$, by unique continuation we conclude that $u_{\infty}\equiv 0$ in $B_{2}$, against the assumption $H_{u_{\infty}}(0,1)=\lim_{k}H_{u_{k}}(0,1)=1$.

The second inequality in (3.1) is proven by the same argument. Indeed, under the same assumption $H_{u}(0,1)=1$, considering that $0\in\Gamma(u)$, we have that $D_{u}(0,1)=I_{u}(0,1)\in[1+s,A]$. Therefore, given a sequence $u_{k}$ contradicting the claim, we deduce the existence of a solution $u_{\infty}$ such that $0=D_{u_{\infty}}(0,1)=\lim_{k}D_{u_{\infty}}(0,1)\in[1+s,A]$, which is impossible.

Finally, (3.2) follows straightforwardly from (3.1):

[TABLE]

∎

Lemma 3.2**.**

For every $A>0$ there exists $C_{\ref{l:monotonia}}=C_{\ref{l:monotonia}}(A)>0$ such that, if $u$ is a solution to the thin obstacle problem (1.1) in $B_{2r_{1}}(x_{0})\subset\mathbb{R}^{n+1}$ with $x_{0}\in\Gamma(u)$ and $I_{u}(x_{0},2r_{1})\leq A$, then for every $r_{0}\in(\nicefrac{{r_{1}}}{{8}},r_{1})$

[TABLE]

Proof.

By rescaling, it suffices to prove the lemma for $x_{0}=0$ and $r_{1}=1$. We start off with the following computation:

[TABLE]

We now use the following integral estimate (whose elementary proof is left to the readers)

[TABLE]

in order to deduce

[TABLE]

Now recall that by (2.13) we have that $H_{u}(t)\leq H_{u}(2)$ for all $t\leq 2$. Hence, from (3.6) we get

[TABLE]

where we used that $r_{0}\geq\frac{1}{8}$ and and $I_{u}(2)\leq A$. ∎

3.1. Oscillation estimate of the frequency

We introduce the following notation for the radial variation of the frequency at a point $x\in\Gamma(u)$: given $0<\rho<r$, we set

[TABLE]

The following lemma shows how the spatial oscillation of the frequency in two nearby points at a given scale is in turn controlled by the radial variations at comparable scales. Here, we exploit for the thin obstacle problem an argument introduced in [16, Theorem 4.2] for multiple-valued functions.

Proposition 3.3**.**

For every $A>0$ there exists $C_{\ref{p:D_x frequency}}(A)>0$ such that, if $\rho>0$, $R>6$ and $u:B_{4R\rho}(x_{0})\to\mathbb{R}$ is a solution to the thin obstacle problem (1.1) in $B_{4R\rho}$, with $x_{0}\in\Gamma(u)$ and $I_{u}(x_{0},4R\rho)\leq A$, then

[TABLE]

for every $x_{1},x_{2}\in B^{\prime}_{\rho}$.

Proof.

1. Without loss of generality, we show the proposition for $x_{0}=0$ and $\rho=1$. The proof is based on estimating the tangential derivative of the frequency function $x\mapsto I_{u}(x,t)$ for a fixed radius $t>0$. Thus, we start off noticing that the functions $x\mapsto H_{u}(x,t)$ and $x\mapsto D_{u}(x,t)$ are differentiable and, for every $e\in\mathbb{R}^{n+1}$ with $e\cdot e_{n+1}=0$, we have that

[TABLE]

and

[TABLE]

where the second equality follows from the divergence theorem applied to the vector field V(y):=\phi\big{(}\textstyle{\frac{|y|}{t}}\big{)}\,\partial_{e}u(y+x)\,|y_{n+1}|^{a}\nabla u(y+x) (note that $V\in C^{0}_{c}(B_{t}(x))$ and by Theorem 2.1 and by Corollary 2.4 the divergence of $V$ does not concentrate on $B_{1}^{\prime}$). We consider next $e:=x_{2}-x_{1}$, and set

[TABLE]

[TABLE]

Then, we have that $\partial_{e}u(z)=\Delta I\cdot u(z)+\Delta\mathcal{E}(z)$ and from (2.8), (3.8) – (3.1) we get also

[TABLE]

and

[TABLE]

In particular, by direct computation

[TABLE]

2. We use now (3.1) with $t=R$ and $x\in B_{1}^{\prime}$. Note that, since $x\in B_{1}^{\prime}$, by (2.1) – (2.3), (2.14) and (3.1) we infer that

[TABLE]

for some constant $C=C(A)>0$. Hence, we have that

[TABLE]

In order to estimate the integral term in (3.11), we notice that

[TABLE]

therefore

[TABLE]

where we used $R>6$ and a direct computation to estimate

[TABLE]

for a dimensional constant $C>0$. We are in the position to apply Lemma 3.2:

[TABLE]

Using (3.11) – (3.13), we get

[TABLE]

having used (2.11) and (3.1) to infer that

[TABLE]

In this respect, recall that $x,x_{i}\in B_{1}^{\prime}$ and $R>6$, so that we are in the position to apply Lemma 3.1. The conclusion now follows by integrating (3.14) along the segment $\{x_{1}+r\,e:\,r\in[0,1]\}$. ∎

4. Mean-flatness and frequency function

4.1. Mean-flatness

We are going to use the following generalization of the Jones’ $\beta$-numbers introduced in [27], also called mean-flatness, which have been already extensively used in the literature (cf., for example, [2, 6, 16, 31, 32] and the list of references therein). We adopt the standard notation ${\textup{dist}}(y,E):=\inf_{x\in E}|y-x|$ for the distance of a point $y$ from a given subset $E\subset\mathbb{R}^{n+1}$.

Definition 4.1**.**

Given a Radon measure $\mu$ in $\mathbb{R}^{n+1}$, $p\in[1,+\infty)$ and $k\in\{0,1,\ldots,n\}$, for every $x_{0}\in\mathbb{R}^{n}$ and for every $r>0$, we set

[TABLE]

where the infimum is taken among all affine $k$-dimensional planes ${\mathcal{L}}\subset\mathbb{R}^{n+1}$.

In the case $p=2$ we have the following elementary characterization. Let $x_{0}\in\mathbb{R}^{n+1}$ and $r>0$ be such that $\mu(B_{r}(x_{0}))>0$, and let us denote by $\bar{x}_{x_{0},r}$ the barycenter of $\mu$ in $B_{r}(x_{0})$, i.e.

[TABLE]

Let moreover ${\bf B}_{x_{0}}:\mathbb{R}^{n+1}\times\mathbb{R}^{n+1}\to\mathbb{R}$ be the symmetric positive semi-definite bilinear form given by

[TABLE]

By standard linear algebra there exists an orthonormal basis of vectors in $\mathbb{R}^{n+1}$ which diagonalizes the bilinear form ${\bf B}_{x_{0}}$: namely, there exist $\{v_{1},\ldots,v_{n+1}\}\subset\mathbb{R}^{n+1}$ (in general not uniquely determined) such that

• (i)

$\{v_{1},\ldots,v_{n+1}\}$ is a Euclidean orthonormal basis, i.e. $v_{i}\cdot v_{j}=\delta_{ij}$;

• (ii)

${\bf B}_{x_{0}}(v_{i},v_{i})=\lambda_{i}$, for some $0\leq\lambda_{n+1}\leq\lambda_{n}\leq\cdots\leq\lambda_{1}$;

• (iii)

${\bf B}_{x_{0}}(v_{i},v_{j})=0$ for $i\neq j$.

The characterization is then the following: the infimum in the definition of $\beta^{(k)}_{\mu,2}$ is reached by all the affine planes ${\mathcal{L}}=\bar{x}_{x_{0},r}+\textup{Span}\{v_{1},\ldots,v_{k}\}$ and

[TABLE]

[TABLE]

In the ensuing sections we are going to consider only the case $k=n-1$ and $p=2$: in order to simplify the notation we will always write $\beta_{\mu}$ for $\beta^{(n-1)}_{\mu,2}$.

4.2. Control of the mean-flatness via the frequency

The main link between Jones’ $\beta$-numbers and the geometric properties of the free boundary is given by the following observation: the mean-flatness of an arbitrary measure $\mu$ supported on $\Gamma(u)$ is controlled by the integration with respect to $\mu$ of suitable radial oscillations of the frequency. This follows closely the approach by Naber–Valtorta [31, Theorem 7.1] for harmonic maps and minimal surfaces. Because of the intrinsic renormalization of the frequency function here we need to use the estimate in Proposition 3.3 as done in [16] for multiple-valued functions.

Proposition 4.2**.**

For every $A>0$ and $R>5$, there exists a constant $C_{\ref{p:mean-flatness vs freq}}(A,R)>0$ with this property. Let $r>0$, $u\in H^{1}(B_{(4R+10)r}(x_{0}),|x_{n+1}|^{a}{\mathcal{L}}^{n+1})$ be a solution to the thin obstacle problem (1.1) in $B_{(4R+10)r}(x_{0})$, with $x_{0}\in\Gamma(u)$ and $I(x_{0},(4R+10)r)\leq A$, and let $\mu$ be a finite Borel measure with ${\rm spt}\,(\mu)\subseteq\Gamma(u)$; then

[TABLE]

Proof.

1. We can assume without loss of generality that $x_{0}=0$ and that $p\in\Gamma(u)\cap B_{r}$ is such that $\mu(B_{r}(p))>0$ (otherwise, there is nothing to prove). Let $\bar{x}=\bar{x}_{p,r}$ be the barycenter of $\mu$ in $B_{r}(p)$ and let $\{v_{1},\ldots,v_{n+1}\}$ be any diagonalizing basis for the bilinear form ${\bf B}_{p}$ introduced in § 4.1, with corresponding eigenvalues $0\leq\lambda_{n+1}\leq\lambda_{n}\leq\cdots\leq\lambda_{1}$. Note that, since by assumption ${\rm spt}\,(\mu)\subset\Gamma(u)\subset\mathbb{R}^{n}\times\{0\}$, we can assume without loss of generality that $v_{n+1}=e_{n+1}$, $\lambda_{n+1}=0$ and hence $\beta_{\mu}(p,r)=(r^{-n-1}\lambda_{n})^{\nicefrac{{1}}{{2}}}$ by (4.3). Therefore, without loss of generality we may also assume that $\lambda_{n}>0$.

From (4.2) and the definition of barycenter we deduce that, for every $i\in\{1,\ldots,n\}$, for every $z\in B_{(2R+5)r}$ and for every constant $\alpha\in\mathbb{R}$, we have

[TABLE]

For the rest of the proof we set

[TABLE]

Using Hölder inequality we deduce that

[TABLE]

Denoting with $\nabla_{\tau}u=(\partial_{1}u,\ldots,\partial_{n}u)$ the tangential gradient, and recalling that

[TABLE]

by integrating over $B_{(R+1)r}(p)\setminus B_{Rr}(p)$ the previous inequality with respect to the measure $|z_{n+1}|^{a}{\mathcal{L}}^{n+1}$ we get

[TABLE]

Next we estimate the two sides of (4.2).

2. For what concerns the l.h.s. of (4.2), we claim the following: there exists a constant $C=C(A,R)>0$ such that

[TABLE]

We argue by contradiction. If the claim were not true, after a suitable rescaling replacing $u$ with $u_{p,r}$, we could find a sequence of solutions $u_{k}$ to the thin obstacle problem in $B_{2R+4}$ with $0\in\Gamma(u_{k})$, such that $I_{u_{k}}(R+3)\leq A^{\prime}:=A+C_{\ref{l:lim uniforme}}(A)$, $H_{u_{k}}(R+2)=1$ and

[TABLE]

(recall that $B_{(2R+4)r}(p)\subset B_{(4R+10)r}$ and by Lemma 3.1 we have $I_{u}(p,(R+3)r)\leq A^{\prime}$). By Corollary 2.8 we have that $H_{u_{k}}(R+3)\leq\left(\nicefrac{{(R+3)}}{{(R+2)}}\right)^{n+a+2A^{\prime}}$ and hence by Corollary 2.10, (up to subsequences, not relabeled) $u_{k}$ converge in $H^{1}(B_{R+2},|x_{n+1}|^{a}{\mathcal{L}}^{n+1})$ to a solution $u_{0}$ to the thin obstacle problem in $B_{R+2}$ with $H_{u_{0}}(R+2)=1$ and

[TABLE]

We deduce from the latter equality that $u_{0}$ depends only on the variable $x_{n+1}$ (recall that $u_{0}$ is analytic in $B_{1}^{+}$). In particular, $u_{0}(x)=-c\,|x_{n+1}|^{2s}$ for some $c>0$, and

[TABLE]

where we used Lemma 2.9. This contradicts $\lim_{k}I_{u_{k}}(t)=I_{u_{0}}(t)$ and concludes the proof of (4.7).

3. Now we estimate the r.h.s. of (4.2) from above. By the triangular inequality we have that

[TABLE]

For every $x\in{\rm spt}\,(\mu)\cap B_{r}(p)$, (3.2) in Lemma 3.1 yields

[TABLE]

since $B_{r}(p)\subseteq B_{2r}$ and $u$ is defined on $B_{(2R+4)r}(p)\subset B_{(4R+10)r}$. By using Lemma 3.2, we can estimate the first integral above as follows:

[TABLE]

in the last inequality we have used Lemma 3.1 (because $|p-x|<r$ and $u$ is defined in $B_{(4R+8)r}(p)\subset B_{(4R+10)r}$). On the other hand, using Jensen’s inequality and Proposition 3.3 (recall that ${\rm spt}\,(\mu)\subseteq\Gamma(u)$), we deduce that

[TABLE]

Finally, note that

[TABLE]

where once again in the last inequality we have used Lemma 3.1.

4. We can now collect the estimates (4.7) – (4.10) to get

[TABLE]

From $I_{u}(p,(R+2)r)\geq 1+s$ (cf. Corollary 2.12), one can then infer (4.4). ∎

5. Homogeneous and almost homogeneous solutions

For the proof of the main theorems, we need to discuss some results concerning homogeneous and almost homogeneous solutions to the thin obstacle problem (1.1).

5.1. Spines of homogeneous solutions

We denote by ${\mathcal{H}}_{\lambda}$ the space of all (non-trivial) $\lambda$-homogeneous solutions to the thin obstacle problem (1.1),

[TABLE]

and set ${\mathcal{H}}:=\bigcup_{\lambda\geq 1+s}{\mathcal{H}}_{\lambda}$. The restriction $\lambda\geq 1+s$ is imposed in view of Corollary 2.12. Recall next the definition of spine of homogeneous solutions (see, e.g., [19]).

Definition 5.1**.**

Let $u\in{\mathcal{H}}$ be a homogeneous solution. The spine $S(u)$ is the maximal subspace of invariance of $u$:

[TABLE]

Simple characterizations of the spine are provided in the next result.

Lemma 5.2**.**

Let $u\in{\mathcal{H}}$ be given. The following are equivalent conditions:

• (i)

$x\in S(u)$,

• (ii)

$u$* is homogeneous with respect to $x$, i.e. $I_{u}(x,r)=I_{u}(x,0^{+})$ for all $r>0$,*

• (iii)

$I_{u}(x,0^{+})=I_{u}(0,0^{+})$.

Proof.

The very definition of spine yields straightforwardly that (i) implies (ii) and (iii). To see that (ii) implies (iii), we consider the functions $u_{0,r_{k}}$ as defined in (2.23), for a sequence of radii $r_{k}\uparrow+\infty$ such that $u_{0,r_{k}}$ converge to some $u_{\infty}$ in $H^{1}_{\textup{loc}}(\mathbb{R}^{n+1},|x_{n+1}|^{a}{\mathcal{L}}^{n+1})$. Then, by Remark 2.6 we infer that

[TABLE]

Similarly, (iii) implies (ii): let $r_{k}\uparrow+\infty$ be a sequence as above, then using $u\in{\mathcal{H}}$ we get

[TABLE]

In particular, taking into account the monotonicity of the frequency, we infer that $I_{u}(x,r)=I_{u}(x,0^{+})$ for every $r>0$, i.e. (ii). Finally, we are left to show that (ii) and (iii) imply (i). By (ii) and (iii) we have that

[TABLE]

with $\lambda=I_{u}(0,0^{+})$. Hence, for every $y\in\mathbb{R}^{n}\times\{0\}$ we have

[TABLE]

5.2. Classification of solutions with maximal dimension of the

spine

There are no homogeneous functions $u\in{\mathcal{H}}$ with spine of dimension $n$, because the only non-trivial solutions of (1.1) even with respect to $x_{n+1}$ and depending only on the variable $x_{n+1}$ are of the form $c\,|x_{n+1}|^{2s}$ with $c<0$. The latter functions have homogeneity $2s<1+s$, that is not allowed for functions in ${\mathcal{H}}$. Therefore, the maximal dimension of the spine of a function in ${\mathcal{H}}$ is at most $n-1$. We say that $u\in{\mathcal{H}}^{\textup{top}}$ if $u\in{\mathcal{H}}$ and $\dim S(u)=n-1$, and we set ${\mathcal{H}}^{\textup{low}}:={\mathcal{H}}\setminus{\mathcal{H}}^{\textup{top}}$ otherwise. All functions in ${\mathcal{H}}^{\textup{top}}$ are classified in the following list.

Lemma 5.3**.**

$u\in{\mathcal{H}}^{\textup{top}}$* if and only if there exists a uniquely determined $\lambda$-homogeneous function $h_{\lambda}:\mathbb{R}^{2}\to\mathbb{R}$, with $\lambda\in\{2m,2m-1+s,2m+2s\}_{m\in{\mathbb{N}}\setminus\{0\}}$, such that*

[TABLE]

for some $c>0$ and $e\in\mathbb{R}^{n}\times\{0\}$ with $|e|=1$. In particular, if $u\in{\mathcal{H}}^{\textup{top}}$ then $\mathcal{N}(u)=S(u)$, and more precisely: if $u(x)=c\,h_{\lambda}(x\cdot e,x_{n+1})$, then

• (I)

if $\lambda=2m$: \Lambda(u)=\Gamma(u)=\mathcal{N}(u)=S(u)=\big{\{}x\cdot e=x_{n+1}=0\big{\}};

• (II)

if $\lambda=2m-1+s$: \Lambda(u)=\big{\{}x\cdot e\leq 0,\,x_{n+1}=0\big{\}} and \Gamma(u)=\mathcal{N}(u)=S(u)=\big{\{}x\cdot e=x_{n+1}=0\big{\}};

• (III)

if $\lambda=2m+2s$: \Lambda(u)=\big{\{}x_{n+1}=0\big{\}}, $\Gamma(u)=\emptyset$ and \mathcal{N}(u)=S(u)=\big{\{}x\cdot e=x_{n+1}=0\big{\}}.

The proof of Lemma 5.3 is a consequence of the full characterization of the homogeneous solutions $v:\mathbb{R}^{2}\to\mathbb{R}$ to the thin obstacle problem. Introducing polar co-ordinates $v(\rho\cos\theta,\rho\sin\theta)=\rho^{\lambda}\,y(\theta)$ with $y:[0,\pi]\to\mathbb{R}$, the system (1.1) can be written in the form:

[TABLE]

with the following four possible boundary conditions:

[TABLE]

The four cases (5.2) – (5.5) determine the corresponding exponents $\lambda$ and the solutions $h_{\lambda}$ as in (I), (II), (III) of the lemma. The proof in general requires the use of the associated Legendre functions and is postponed to the appendix where we establish also other complementary results that are mandatory for the analysis in Section 8 (cf. Proposition A.1). Here we give the details for the simplest case of the Signorini problem $s=\nicefrac{{1}}{{2}}$, i.e. $a=0$.

Proof of Lemma 5.3 for $s=\nicefrac{{1}}{{2}}$..

If $a=0$, the general integral of (5.1) is

[TABLE]

with $A_{1},A_{2}\in\mathbb{R}$. We can then discuss the four possible cases.

• (I)

For (5.2) we have that $y^{\prime}(0)=0$ implies $A_{2}=0$ and $A_{1}\neq 0$, and $y^{\prime}(\pi)=0$ gives $\lambda\in{\mathbb{N}}$. Considering that $y(0)>0$ we find $A_{1}>0$, and by $y(\pi)>0$ one gets $\lambda=2m$ with $m\in{\mathbb{N}}\setminus\{0\}$.

• (II)

For (5.3), we have that $y(0)=0$ gives $A_{1}=0$ and $A_{2}\neq 0$. In turn $y^{\prime}(0)\leq 0$ implies $A_{2}<0$. Thus, $y^{\prime}(\pi)=0$ yields $\cos(\lambda\pi)=0$, i.e. $\lambda=m+\nicefrac{{1}}{{2}}$ for $m\in{\mathbb{N}}\setminus\{0\}$, and finally $m$ is odd since $y(\pi)>0$. One proceeds analogously in case (5.4).

• (III)

Finally, for (5.5), we have that $y(0)=y(\pi)=0$ implies $A_{1}=0$, $A_{2}\neq 0$ and $\lambda\in{\mathbb{N}}$. Considering that $y^{\prime}(0)\leq 0\leq y^{\prime}(\pi)$ we conclude that $A_{2}<0$ and $\lambda$ odd.

In all the cases the nonzero coefficient $A_{i}$ is chosen suitably in order to satisfy the normalization condition $H_{h_{\lambda}}(1)=1$. The statements concerning $\Gamma(u)$, $\Lambda(u)$, $\mathcal{N}(u)$ and $S(u)$ are now direct consequences of the explicit formulas for the solutions. ∎

For the lowest frequency $1+s$, actually all homogeneous solutions have maximal spine as proved by Caffarelli, Salsa and Silvestre in [12], this result can be equivalently stated by the inclusion

[TABLE]

5.3. Almost homogeneous solutions

We next introduce the notion of almost homogeneous solutions.

Definition 5.4**.**

Let $\eta>0$ and $R>1$ be given constants. A solution $u:B_{R}\to\mathbb{R}$ to thin obstacle problem (1.1) is called $\eta$-almost homogeneous if $0\in\Gamma(u)$ and

[TABLE]

The following lemma justifies this terminology.

Lemma 5.5**.**

For every $\delta,A>0$ and $R>1$ there exists $\eta>0$ with the following property: let $u$ be a $\eta$-almost homogeneous solution with $I_{u}(R)\leq A$ and $H_{u}(R)=1$; then, there exists a homogeneous solution $w\in{\mathcal{H}}$ such that

[TABLE]

Proof.

We argue by contradiction: assume there exist $\delta,A,R$ as in the statement and a sequence $(u_{k})_{k\in{\mathbb{N}}}$ of $k^{-1}$-homogeneous solutions in $B_{R}$ with $I_{u_{k}}(R)\leq A$ such that

[TABLE]

We can then apply Corollary 2.10 and find a subsequence (not relabeled) and a solution $u_{0}$ to the obstacle problem in $B_{R}$ such that $u_{k}\to u_{0}$ in $H^{1}(B_{R-1},|x_{n+1}|^{a}\,{\mathcal{L}}^{n+1})$. Note that

[TABLE]

In particular, $u_{0}$ is non-trivial and

[TABLE]

By Proposition 2.7 we infer that $u_{0}$ is homogeneous of degree $I_{u_{0}}(1)=\lim_{k\uparrow+\infty}I_{u_{k}}(1)\geq 1+s$, because $0\in\Gamma(u_{k})$. Therefore, $u_{0}\in{\mathcal{H}}$ and this contradicts (5.8). ∎

Next we show a rigidity result which will be used crucially in the estimate of the Hausdorff measure of the free boundary.

Proposition 5.6**.**

For every $\tau,A>0$ there exists $\eta_{\ref{p:rigidity}}(\tau,A)>0$ with this property. Let $u:B_{4}\to\mathbb{R}$ be a $\eta_{\ref{p:rigidity}}$-almost homogeneous solution to the thin obstacle problem with $I_{u}(0,4)\leq A$. Then, the following dichotomy holds:

• (i)

either for every point $x\in\Gamma(u)\cap B_{2}$ we have

[TABLE]

• (ii)

or there exists a linear subspace $V\subset\mathbb{R}^{n}\times\{0\}$ of dimension $n-2$ such that

[TABLE]

recall the notation ${\mathcal{T}}_{\tau}(V):=\{x\in\mathbb{R}^{n+1}:{\textup{dist}}(x,V)<\tau\}$.

Proof.

The proof is by contradiction. We assume that there exist $\tau,A$ as in the statement and a sequence $(u_{k})_{k\in{\mathbb{N}}}$ of $k^{-1}$-almost homogeneous solutions in $B_{4}$ with $I_{u_{k}}(4)\leq A$ contradicting the statement, i.e. both (i) and (ii) simultaneously fail: namely, there exists $x_{k}\in\Gamma(u_{k})\cap B_{2}$ such that

[TABLE]

and for every linear subspace $V\in\mathbb{R}^{n}\times\{0\}$ of dimension $n-2$ there exists $y_{k}\in\Gamma(u_{k})\cap B_{2}$ (a priori depending on $V$) such that

[TABLE]

By eventually rescaling the functions of the sequence, we can assume without loss of generality that $H_{u_{k}}(0,4)=1$. In particular, it follows from Lemma 5.5 that

[TABLE]

Up to passing to a subsequence (not relabeled) we distinguish to cases:

• (a)

either there exists $w_{k}\in{\mathcal{H}}^{\textup{top}}$ such that $\lim_{k}\|u_{k}-w_{k}\|_{H^{1}(B_{3},|x_{n+1}|^{a}\,{\mathcal{L}}^{n+1})}=0$;

• (b)

or there exists $\delta>0$ such that

[TABLE]

In case (a) we show that (5.11) cannot hold. Indeed, by Corollary 2.10 there exist a homogeneous function $w_{0}\in{\mathcal{H}}^{\textup{top}}$ (note that ${\mathcal{H}}^{\textup{top}}$ is closed under locally strong $H^{1}$ convergence), a point $x_{0}\in\bar{B}_{2}$ and a subsequence (not relabeled) such that

[TABLE]

In particular,

[TABLE]

Note that $w_{0}\not\equiv 0$, because $H_{w_{0}}(0,3)=\lim_{k}H_{u_{k}}(0,3)\geq(\nicefrac{{3}}{{4}})^{n+a+2A}$ thanks to (2.12) in Corollary 2.8. This implies that $I_{w_{0}}(x_{0},1)=I_{w_{0}}(0,1)$, since $x_{0}\in\mathcal{N}(w_{0})=S(w_{0})$ being $w_{0}\in{\mathcal{H}}^{\textup{top}}$ and Lemma 5.3, which gives the desired contradiction.

In case (b), by combining (5.13) and (5.14), and by the compactness in Corollary 2.10 (up to passing to a subsequence, not relabeled) there exists $v_{0}\in{\mathcal{H}}$ such that

[TABLE]

Moreover, from (5.14) we deduce that $v_{0}\in{\mathcal{H}}^{\textup{low}}$ (note that $v_{0}\not\equiv 0$ because we have that $H_{v_{0}}(0,3)=\lim_{k}H_{u_{k}}(0,3)\geq(\nicefrac{{3}}{{4}})^{n+a+2A}$ by Corollary 2.8). We now show that we have a contradiction to (5.12) with $V$ any $(n-2)$-dimensional subspace containing $S(v_{0})$. Indeed, let $y_{k}$ be as in (5.12) for such a choice of $V$; by compactness, up to passing to a subsequence (not relabeled), there exists $y_{0}\in\bar{B}_{2}$ such that

[TABLE]

Proposition 2.7 implies that $I_{v_{0}}(y_{0},r)=I_{v_{0}}(y_{0},0^{+})$ for every $r>0$ and by Lemma 5.2 we must have $y_{0}\in S(v_{0})$, thus contradicting $S(v_{0})\subseteq V$ and $y_{0}\not\in{\mathcal{T}}_{\tau}(V)$. ∎

6. The measure of the free boundary

In this section we prove Theorem 1.1 that provides a local estimate of the Minkowski content, and thus of the Hausdorff measure, of the free boundary in the lower dimensional obstacle problem. Here we use a modified version of the inductive covering argument in [31, Section 8]. The key monotone quantity we consider is the maximal function of the frequency

[TABLE]

Theorem 1.1 is a direct consequence of the following proposition.

Proposition 6.1**.**

For every $L>0$, there exists a constant $C_{\ref{p:induzione}}(L)>0$ with this property: for any solution $u\not\equiv 0$ to the thin obstacle problem (1.1) in $B_{2\rho}(z)\subset\mathbb{R}^{n+1}$ with $z\in\mathbb{R}^{n}\times\{0\}$, we have

[TABLE]

Proof of Theorem 1.1.

We are given $u$ a solution to the lower dimensional obstacle problem in $B_{1}$ and $K\subset\subset B_{1}$. Set $\delta:=4^{-1}{\textup{dist}}(K,\partial B_{1})$, let $\{B_{\delta}(x_{i})\}_{i\in J}$, with $J$ a (finite) maximal subset of points in $\Gamma(u)\cap K$, having mutual distance at least $\delta$. Set $L:=\max_{i\in J}\Theta_{u}(x_{i},2\delta)$. Then, by applying Proposition 6.1 to every $B_{2\delta}(x_{i})$, we have that

[TABLE]

We point out that the constant $C$ depends only $n$, on $I_{u}(1)$ and on ${\textup{dist}}(K,\partial B_{1})$. Indeed, $L$ depends on $I_{u}(1)$ via Lemma 3.1; and since the balls $B_{\nicefrac{{\delta}}{{2}}}(x_{i})$ are disjoint, contained in $B_{1}$ and with centers in $B_{1}^{\prime}$, we can estimate ${\mathcal{H}}^{0}(J)\leq(\nicefrac{{2}}{{\delta}})^{n}$. ∎

The rest of the section is devoted to the proof of Proposition 6.1.

6.1. Proof of Proposition 6.1

By rescaling it is enough to consider the case $z=0$ and $\rho=1$. We start off with the case of minimal frequency $L=1+s$: then, by Corollary 2.12 $\Theta_{u}(0,1)=1+s$ and thus $u\in{\mathcal{H}}_{1+s}$ (cf. (5.6)). In turn, this implies that $\Gamma(u)$ is a $(n-1)$-dimensional hyperplane of $\{x_{n+1}=0\}$ and (6.2) follows at once.

The proof is then completed by showing that $\sup{\mathcal{S}}=+\infty$ where

[TABLE]

The latter claim is in turn implied by the following fact: for every $L_{0}>1+s$ there exists a constant $\eta(L_{0})>0$ such that, if $L\in{\mathcal{S}}$ and $L<L_{0}$ then $L+\eta(L_{0})\in{\mathcal{S}}$. In order to specify $\eta(L_{0})$ we need to introduce several dimensional constants; to show the consistency of their choices, we declare them at the beginning (the readers can skip this list and refer to it each time the constants are introduced):

• •

$C_{\ref{l:misura 2}}:=10^{3(n-1)}C_{\ref{t:Reif discreto}}(n)$, where $C_{\ref{t:Reif discreto}}(n)>1$ is the dimensional constant of Theorem 6.3;

• •

$\lambda=\min\{10^{-3},16^{2-n}\,C_{\ref{l:misura 2}}^{-1}\}$;

• •

\tau=\min\big{\{}\lambda^{2},10^{-20n}\,C_{\ref{p:mean-flatness vs freq}}(2L_{0},45)^{-2}\,C_{\ref{l:misura 2}}^{-2}\,\lambda^{4n}\,\delta_{\ref{t:Reif discreto}}^{2}(\lambda)\big{\}}, where $C_{\ref{p:mean-flatness vs freq}}(2L_{0},45)$ is the constant in Proposition 4.2 corresponding to $R=45$ and $A=2L_{0}$, and $\delta_{\ref{t:Reif discreto}}(\lambda)$ is the constant introduced in Theorem 6.3;

• •

0<\eta\leq\min\big{\{}\eta_{\ref{p:rigidity}}(\tau,2L_{0}),\tau,L_{0}\big{\}}, where $\eta_{\ref{p:rigidity}}(\tau,2L_{0})$ is the constant introduced in Proposition 5.6 with parameters, $\tau$ and $2L_{0}$.

Note that, for ever $L<L_{0}$ we have that $L+\eta\leq 2L_{0}$ with such a choice of $\eta$.

Then, the proof of Proposition 6.1 consists in showing that (6.2) holds for $L+\eta$, supposing that it has been verified for $L<L_{0}$. We proceed in several steps.

1. Let $u$ be a solution in $B_{2}$ of the lower dimensional obstacle problem with $\Theta_{u}(0,1)\leq L+\eta$, and let $r\in(0,1)$ be the size of the tubular neighborhood in (6.2) (recall that $\rho=1$ by scaling). For every $x\in\Gamma(u)\cap B_{\nicefrac{{1}}{{2}}}$ we set s_{x}:=\max\big{\{}r_{x},2r\big{\}} with

[TABLE]

By definition, if $I_{u}(x,1)>L$, then

[TABLE]

Let now $\{x_{i}\}_{i\in J}\subset\Gamma(u)\cap B_{\nicefrac{{1}}{{2}}}$ be a finite collection of points such that the balls $B_{\nicefrac{{s_{x_{i}}}}{{2}}}(x_{i})$ constitute a Vitali covering of $\Gamma(u)\cap B_{\nicefrac{{1}}{{2}}}$: i.e.

[TABLE]

By construction, we have that

• (i)

{\mathcal{T}}_{r}\big{(}\Gamma(u)\cap B_{\nicefrac{{s_{x_{i}}}}{{2}}}(x_{i})\big{)}\subseteq B_{s_{x_{i}}}(x_{i}), for all $i\in J$ because $2r\leq s_{x_{i}}$,

• (ii)

$\Theta_{u}(x_{i},s_{x_{i}})=L$ if $s_{x_{i}}>2r$.

The key estimate is to show that there exists a dimensional constant $\bar{C}>0$ such that

[TABLE]

Indeed, assuming momentarily (6.5) we can prove (6.2) for $L+\eta$ as follows:

[TABLE]

with $C_{\ref{p:induzione}}(L+\eta):=\left(C_{\ref{p:induzione}}(L)+2^{n+1}\omega_{n+1}\right)\,\bar{C}$. In the third inequality, we have used (6.2) itself with bound $L$ on $\Theta_{u}(x_{i},s_{x_{i}})$ in view of (ii).

2. Next we want to prove the claim (6.5). Let $\lambda$ be the constant introduced at the beginning, we consider a suitable decomposition of the sets of centers $\{x_{i}\}_{i\in J}$:

[TABLE]

with A^{(0)}:=\big{\{}x_{i},\,i\in J\,:\,s_{x_{i}}\geq\lambda^{2}\big{\}}, $N(\lambda)=\lfloor 10^{n}\lambda^{-3n}\rfloor+1$ and $A^{(l)}$ satisfying the following condition for $l>0$:

[TABLE]

To see that such a decomposition exists, we follow [31, Lemma 5.4] and proceed inductively. We order the points in $A\setminus A^{(0)}$ according to a decreasing order of the corresponding radii: i.e., $A\setminus A^{(0)}=\{p_{i}\}_{i\in J^{\prime}}$ with $s_{p_{i+1}}\leq s_{p_{i}}$. Then, $p_{1}\in A^{(1)}$ and, if $p_{1},\ldots,p_{i-1}$ have been sorted out, we assign $p_{i}$ to some $A^{(l)}$ so that $A^{(l)}$ does not contain any point $p_{j}$ with $j\leq i-1$, for which

[TABLE]

Note that for every $j$ satisfying (6.7) we have $|p_{i}-p_{j}|\leq\nicefrac{{s_{p_{j}}}}{{\lambda}}\leq\nicefrac{{s_{p_{i}}}}{{\lambda^{3}}}$, thus $p_{j}\in B_{\nicefrac{{s_{p_{i}}}}{{\lambda^{3}}}}(p_{i})$; moreover, the balls $B_{\nicefrac{{s_{p_{i}}}}{{10}}}(p_{j})$ with $j$ as in (6.7) are disjoint, as $s_{p_{i}}\leq s_{p_{j}}$ and $B_{\nicefrac{{s_{p_{j}}}}{{10}}}(p_{j})$ are disjoint by construction. Therefore, since $N(\lambda)$ is strictly bigger than the number of disjoint balls with radius $\nicefrac{{1}}{{10}}$ in $B_{\nicefrac{{1}}{{\lambda^{3}}}}$ and center on $B_{\nicefrac{{1}}{{\lambda^{3}}}}^{\prime}$, one can surely find $l$ so that $p_{j}\not\in A^{(l)}$ for all $j$ as in (6.7).

Let us check that (6.6) holds. Indeed, if $j<i$ (i.e. $s_{p_{i}}\leq s_{p_{j}}$) and $p_{i}\in B_{\nicefrac{{s_{p_{j}}}}{{\lambda}}}(p_{j})$, then the second condition in (6.7) must fail (being $p_{j}\in A^{(l)}$), i.e. $s_{p_{i}}<\lambda^{2}\,s_{p_{j}}$. On the other hand, if $i<j$ (i.e. $s_{p_{j}}\leq s_{p_{i}}$), from $p_{i}\in B_{\nicefrac{{s_{p_{j}}}}{{\lambda}}}(p_{j})$, we deduce $p_{j}\in B_{\nicefrac{{s_{p_{j}}}}{{\lambda}}}(p_{i})\subset B_{\nicefrac{{s_{p_{i}}}}{{\lambda}}}(p_{i})$ and, as $p_{j}\in A^{(l)}$, the second condition in (6.7) must fail, i.e. $s_{p_{j}}<\lambda^{2}\,s_{p_{i}}$. But this is a contradiction because $\lambda<\nicefrac{{1}}{{10}}$ and

[TABLE]

3. Next, for $l\in\{0,\ldots,N(\lambda)\}$ we introduce the measures:

[TABLE]

To conclude (6.5), we show that there exists a dimensional constant $C_{0}>0$ such that

[TABLE]

Indeed, from (6.9) we infer (6.5) with the constant \bar{C}:=\big{(}N(\lambda)+1\big{)}\,C_{0}:

[TABLE]

The case $l=0$ is straightforward: since the balls $B_{\nicefrac{{\lambda^{2}}}{{10}}}(x)$ with $x\in A^{(0)}$ are pairwise disjoint, contained in $B_{1}$ and with center $x\in B_{\nicefrac{{1}}{{2}}}^{\prime}$, then {\mathcal{H}}^{0}\big{(}A^{(0)}\big{)}\leq\frac{10^{n}}{\lambda^{2n}}. Being $s_{x}\leq 2$ we deduce that

[TABLE]

and estimate (6.9) for $l=0$ follows as soon as $C_{0}\geq\nicefrac{{20^{n}}}{{\lambda^{2n}}}$.

For the remaining cases, we are going to show the following lemma.

Lemma 6.2**.**

Let $\mu^{l}$ be the measures in (6.8) with $l\geq 1$. Then,

[TABLE]

for every $x\in{\rm spt}\,(\mu^{l})$ and for every $\rho\in(s_{x},\lambda^{2}]$, where $C_{\ref{l:misura 2}}>0$ is the dimensional constant introduced at the beginning.

Lemma 6.2 implies (6.9). Indeed, let us consider a maximal subset of points $\{x_{i}\}_{i\in J^{(l)}}\subseteq A^{(l)}$ with $|x_{i}-x_{j}|\geq\lambda^{2}$ for all $i\neq j\in J^{(l)}$. Then, the balls $\{B_{\nicefrac{{\lambda^{2}}}{{2}}}(x_{i})\}_{i\in J^{(l)}}$ are disjoint, contained in $B_{1}$ (as $\lambda<\nicefrac{{1}}{{2}}$), and with centers $x_{i}\in B_{\nicefrac{{1}}{{2}}}^{\prime}$. Thus, ${\mathcal{H}}^{0}(J^{(l)})\leq\nicefrac{{2^{n}}}{{\lambda^{2n}}}$ and by maximality of the number of points in $\{x_{i}\}_{i\in J^{(l)}}$ we have also ${\rm spt}\,(\mu^{l})\subset\cup_{i\in J^{(l)}}B_{\lambda^{2}}(x_{i})$. Then

[TABLE]

and (6.9) follows with C_{0}:=\max\big{\{}\nicefrac{{2^{n}\,C_{\ref{l:misura 2}}}}{{\lambda^{2}}},\nicefrac{{20^{n}}}{{\lambda^{2n}}}\big{\}}.

Proposition 6.1, and hence Theorem 1.1, are now established once we show Lemma 6.2.

6.2. Proof of Lemma 6.2

We fix $l\in\{1,\ldots,N(\lambda)\}$, and set s_{\min}:=\min\big{\{}s_{w}\;:\;w\in A^{(l)}\}, $k_{\max}:=\lfloor\log_{\lambda}(s_{\min})\rfloor$. Note that $k_{\max}\geq 2$ and $\lambda^{k_{\max}+1}<s_{\min}\leq\lambda^{k_{\max}}$. We prove (6.10) for all $\rho\in(\max\{\lambda^{k+1},s_{\min}\},\lambda^{k}]$ by induction on $k\in\{2,\ldots,k_{\max}\}$ in decreasing order. More precisely, the base induction step is for $k=k_{\max}$. In this case, for every point $w_{0}\in{\rm spt}\,(\mu^{l})=A^{(l)}$ with $s_{w_{0}}\leq\lambda^{k_{\max}}$ we have that ${\rm spt}\,(\mu^{l})\cap B_{\lambda^{k_{\max}}}(w_{0})=\{w_{0}\}$, from which (6.10) readily follows. Indeed, if $w\in B_{\lambda^{k_{\max}}}(w_{0})\cap A^{(l)}$ is different from $w_{0}$, then $w\in B_{\nicefrac{{s_{w_{0}}}}{{\lambda}}}(w_{0})\cap A^{(l)}$ as $s_{w_{0}}\in(\lambda^{k_{\max}+1},\lambda^{k_{\max}}]$ and by (6.6) we reach a contradiction

[TABLE]

We can then proceed inductively: we assume that we have shown (6.10) for every $\rho\in(s_{\min},\lambda^{k+1}]$ for some $k\in\{2,\dots,k_{\max}-1\}$ and for all $w\in{\rm spt}\,(\mu^{l})$ with $s_{w}\leq\lambda^{k+1}$, and then we prove that

[TABLE]

1. Let $t\in(\lambda^{k+1},\lambda^{k}]$ with $k\geq 2$ and $w_{0}\in{\rm spt}\,(\mu^{l})$ be such that $s_{w_{0}}<t$. We set $W:={\rm spt}\,(\mu^{l})\cap B_{t}(w_{0})$ and

[TABLE]

where $\tau$ is the constant introduced at the beginning. Next we order the points in $W^{(1)}$ in such a way that $W^{(1)}=\{p_{h}\}_{h}$ with $s_{p_{h}}\geq s_{p_{h+1}}$; and we define inductively $z_{1}:=y_{p_{1}}$ and for $\alpha\geq 2$

[TABLE]

where the $y_{p}$’s are the points defined in (6.3) (which exist because $p_{h}\in A^{(l)}$ with $l\geq 1$ implies $s_{p_{h}}<\lambda^{2}<1$, i.e. $I_{u}(p_{h},1)>L$). Let $Z$ be the set of the selected points $z_{\alpha}$’s and set $s_{z_{\alpha}}:=s_{p_{m_{\alpha}}}$ (with a slight abuse of notation), $E:=\cup_{z_{\alpha}\in Z}B_{s_{z_{\alpha}}}(z_{\alpha})$ and

[TABLE]

The measure $\mu^{l}_{1}$ satisfies the following five properties:

[TABLE]

The properties (6.13) and (6.14) follows directly from the definition of $\mu^{l}_{1}$. More precisely, for (6.13) recall the choice $\eta\leq\tau$ and that by assumption $\Theta_{u}(0,1)\leq L+\eta$. Therefore, the conclusion follows either by (6.3) if $p\in Z$ or otherwise by the very definition of $W^{(2)}$.

For (6.14) we distinguish three cases:

• (i)

$p,\,p^{\prime}\in Z$. Assume without loss of generality that $s_{p}\leq s_{p^{\prime}}$, then by the selection procedure defining $Z$ itself $p\notin B_{s_{p^{\prime}}}(p^{\prime})$, and thus $s_{p^{\prime}}<|p-p^{\prime}|$;

• (ii)

$p\in Z$, $p^{\prime}\in W^{(2)}\setminus E$. Then $p^{\prime}\notin B_{s_{p}}(p)$ by definition of $E$, so that $s_{p}<|p-p^{\prime}|$. Moreover, if $p=y_{w}$, with $w\in W^{(1)}$, we use (6.4) to infer

[TABLE]

• (iii)

$p,\,p^{\prime}\in W^{(2)}\setminus E$. Since $B_{\frac{s_{p}}{10}}(p)\cap B_{\frac{s_{p^{\prime}}}{10}}(p^{\prime})=\emptyset$ by (6.4), $\max\{s_{p},s_{p^{\prime}}\}<10|p-p^{\prime}|$.

For what concerns (6.15), we notice that for all $w\in W$ by (6.4) we have that $s_{w}<10|w-w_{0}|\leq 10t$, and therefore

[TABLE]

Eq. (6.16) and (6.17) are proven in the next two steps. The proof of (6.11) will then be a consequence of (6.13) – (6.17) only and it will be detailed in step 4.

2. For what concerns (6.16), for every $z_{\alpha}\in Z$ we introduce the sets

[TABLE]

Hence, as by the very definition of $E$

[TABLE]

and by that of $Z$

[TABLE]

then $W=\cup_{z\in Z}W^{z}\cup(W^{(2)}\setminus E)$ and

[TABLE]

We will prove (6.16) by showing that, for every $z_{\alpha}\in Z$, we have

[TABLE]

Indeed, from (6.19) we immediately deduce that

[TABLE]

The key observation to establish (6.19) is the following: let $\bar{w}\in W^{(1)}$ be such that $z_{\alpha}=y_{\bar{w}}$. Then, by definition

[TABLE]

We can then apply Proposition 5.6 in $B_{8s_{\bar{w}}}(z_{\alpha})$ with parameters $\tau$, $2L_{0}$. Indeed, $I_{u}(z_{\alpha},8s_{\bar{w}})\leq\Theta_{u}(0,1)\leq L+\eta\leq 2L_{0}$ (recall that $L<L_{0}$ and we have chosen $\eta\leq L_{0}$). Moreover, as we have imposed $\eta\leq\eta_{\ref{p:rigidity}}(\tau,2L_{0})$, we deduce that the first case of the dichotomy of Proposition 5.6 does not occur: i.e. there exists a $(n-2)$-dimensional affine subspace passing through $z_{\alpha}$ such that

[TABLE]

Eq. (6.20) is the main ingredient of the proof, because it implies that all the points in $W^{z_{\alpha}}$ different from $\bar{w}$ have clustered around a lower dimensional space $V$, namely

[TABLE]

Indeed, consider a generic point $w\in W^{z_{\alpha}}\setminus\{\bar{w}\}$. If $w\in W^{(2)}\cap B_{s_{z_{\alpha}}}(z_{\alpha})$, then $w\in B_{2s_{\bar{w}}}(\bar{w})$ and by (6.6) we have $s_{w}\leq\lambda^{2}s_{\bar{w}}$. In turn this implies that $y_{w}\in B_{s_{w}}(w)\subset B_{2s_{z_{\alpha}}}(z_{\alpha})$ and

[TABLE]

Therefore, by (6.20) we infer that $y_{w}\in{\mathcal{T}}_{2\tau s_{\bar{w}}}(V)$ and, since $\tau\leq{\lambda^{2}}$, also $w\in{\mathcal{T}}_{2\tau s_{\bar{w}}+s_{w}}(V)\subset{\mathcal{T}}_{4\lambda^{2}s_{\bar{w}}}(V)$. On the other hand, if $w\in W^{(1)}$, then by the selection procedure (recall the decreasing order of the radii $s_{z_{j}}$), we have that $s_{w}\leq s_{\bar{w}}$: in particular $w\in B_{s_{\bar{w}}}(y_{w})\subset B_{3s_{\bar{w}}}(\bar{w})$. Therefore, thanks to (6.6) we have also $s_{w}\leq\lambda^{2}s_{\bar{w}}$ and (6.22) holds. By (6.20) $y_{w}\in{\mathcal{T}}_{2\tau s_{\bar{w}}}(V)$ and hence $w\in{\mathcal{T}}_{4\lambda^{2}s_{\bar{w}}}(V)$, thus showing (6.21).

Then the proof of (6.19) follows from an elementary covering argument. Let $Q^{\prime}\subset W^{z_{\alpha}}\setminus\{\bar{w}\}$ be a maximal collection of points such that the balls \big{\{}B_{\nicefrac{{\lambda s_{\bar{w}}}}{{20}}}(p)\}_{p\in Q^{\prime}} are pairwise disjoint: in particular $W^{z_{\alpha}}\setminus\{\bar{w}\}\subset\cup_{p\in Q^{\prime}}B_{\nicefrac{{\lambda s_{\bar{w}}}}{{10}}}(p)$. Let $\pi_{V}:\mathbb{R}^{n}\to V$ be the nearest point projection on $V$ and note that, since $\lambda\leq\nicefrac{{1}}{{160}}$, we have

[TABLE]

where we used that every $p\in W^{z_{\alpha}}$ is contained in $B_{3s_{\bar{w}}}(z_{\alpha})$, and thus $p\in B_{4s_{\bar{w}}}(\bar{w})$. Therefore, $B_{\nicefrac{{\lambda s_{\bar{w}}}}{{40}}}(\pi_{V}(p))\cap V$ are pairwise disjoint for $p\in Q^{\prime}$ and contained in $B_{4s_{\bar{w}}}(\pi_{V}(z_{\alpha}))\cap V$. This allows us to give an estimate on the cardinality of $Q^{\prime}$, namely ${\mathcal{H}}^{0}(Q^{\prime})\leq\nicefrac{{160^{n-2}}}{{\lambda^{n-2}}}$. In proving the latter estimate we have crucially used that $V$ has dimension $n-2$. Now by the inductive hypothesis (6.11) we get (6.19):

[TABLE]

thanks to the choice $\lambda<16^{2-n}C_{\ref{l:misura 2}}^{-1}$. We can apply the inductive hypothesis to $B_{\nicefrac{{\lambda s_{\bar{w}}}}{{10}}}(p)$ since $s_{p}\leq\lambda^{2}s_{\bar{w}}<\nicefrac{{\lambda s_{\bar{w}}}}{{10}}\leq\lambda^{k+1}$ (the first inequality holds thanks to (6.6) because for every $p\in Q^{\prime}$ we have that $p\in B_{4s_{\bar{w}}}(\bar{w})$, and the last one in view of $s_{w}\leq 10\,t\leq 10\,\lambda^{k}$ for every $w\in W$).

3. We show next (6.17). Let $p,\rho$ be as in the statement. For every $q\in{\rm spt}\,(\mu_{1}^{l})\cap B_{\rho}(p)$ let $x_{q}\in{\rm spt}\,(\mu^{l})$ be a point such that $y_{x_{q}}=q$ if $q\notin{\rm spt}\,(\mu^{l})$ and coinciding with $q$ itself otherwise. Then,

[TABLE]

Therefore, for every point $q\in{\rm spt}\,(\mu^{l}_{1})\cap B_{\rho}(p)$ we have that the corresponding point $x_{q}$ belongs to ${\rm spt}\,(\mu^{l})\cap B_{41\rho}(x_{p})$, so that

[TABLE]

The proof of (6.17) is now a consequence of the inductive hypothesis (6.11) and a covering argument. Indeed,

• (i)

if $41\rho\leq\lambda^{k+1}$: we can apply (6.10) directly (since $s_{p}\leq 20\rho$ by assumption), and infer that \mu^{l}\big{(}B_{\rho}(p)\big{)}\leq C_{\ref{l:misura 2}}41^{n-1}\rho^{n-1};

• (ii)

if $41\rho>\lambda^{k+1}$: we cover ${\rm spt}\,(\mu^{l})\cap B_{41\rho}(x_{p})$ with balls $B_{\nicefrac{{\lambda^{k+1}}}{{10}}}(w)$ having centers $w\in{\rm spt}\,(\mu^{l})$ such that half the balls are disjoint. Since $\rho\leq 10^{2}\lambda^{k}$ by assumption (cf. (6.17)) and the centers are in $B_{1}^{\prime}$, the cardinality of the cover can be estimated by $(\nicefrac{{10^{5}}}{{\lambda}})^{n}$. Moreover, $s_{w}\leq 20\cdot 41\rho\leq 10^{5}\,\lambda^{k}$ in view of $w\in B_{41\rho}(x_{p})$ (cf. (6.4)), and $\rho\leq 10^{2}\lambda^{k}$ by assumption (cf. (6.17)) . Hence, in case $s_{w}\leq\lambda^{k+1}$ we can use the inductive hypothesis (6.11) to infer that $\mu^{l}(B_{\nicefrac{{\lambda^{k+1}}}{{10}}}(w))\leq C_{\ref{l:misura 2}}\lambda^{(k+1)(n-1)}$. Otherwise, if $s_{w}>\lambda^{k+1}$, ${\rm spt}\,(\mu^{l})\cap B_{\nicefrac{{\lambda^{k+1}}}{{10}}}(w)=\{w\}$ by (6.4), and thus $\mu^{l}(B_{\nicefrac{{\lambda^{k+1}}}{{10}}}(w))=s_{w}^{n-1}\leq 10^{5(n-1)}\,\lambda^{k(n-1)}$. In conclusion, recalling that $\lambda^{k+1}\leq 41\rho$, we infer that

[TABLE]

4. We are now in the position to infer (6.11) from (6.13)–(6.17), thus concluding the proof of Lemma 6.2. We start off estimating the generalized Jones’ number for $\mu^{l}_{1}$ (for simplicity we omit the subscripts in their notation): for every $\rho\in(0,{44}t]$, with $t\in(\lambda^{k+1},\lambda^{k}]$ by (6.11), and $w\in{\rm spt}\,(\mu^{l}_{1})$, using Proposition 4.2 with parameters $A=2L_{0}$ and $R=45$ (recall that $L_{0}+\eta\leq 2L_{0}$ and do not confuse the radius $R$ there with the one in this proof) we infer

[TABLE]

having used that $s_{z}\leq 20\rho$ if $z\in{\rm spt}\,(\mu^{l}_{1})\cap B_{\rho}(w)$ by (6.14). Integrating (6.23) over $B_{R}({\bar{w}})$ for $\bar{w}\in{\rm spt}\,(\mu_{1}^{l})$, with $R\in(0,44\lambda^{k}]$ and $\rho\in(0,R]$, we get

[TABLE]

In the second inequality we have used Fubini’s theorem, and we have set for simplicity $\bar{C}(\lambda):=C_{\ref{p:mean-flatness vs freq}}\,C_{\ref{l:misura 2}}\,\nicefrac{{10^{12n}}}{{\lambda^{2n}}}$. Let us now introduce the following notation for the average oscillation of a measure $\mu$ at scale $\lambda$ on the ball $B_{\varrho}(\bar{w})$:

[TABLE]

Then, summing (6.2) for $\rho=\lambda^{j}R$ with $j\in{\mathbb{N}}$ and using $\lambda\leq 10^{-3}$, we get

[TABLE]

by taking into account that $\nicefrac{{s_{z}}}{{20}}\leq\rho<R$ and $20\lambda^{-1}R<1$ (being $R\leq 44\lambda^{k}$ and $k\geq 2$). In addition, we notice that in case $R<\nicefrac{{s_{\bar{w}}}}{{20}}$, estimate (6.2) still holds true. Indeed, in such a case $B_{R}(\bar{w})\cap{\rm spt}\,(\mu^{l}_{1})=\{\bar{w}\}$ and by definition $\beta(\bar{w},\lambda^{j}\,R)=0$ for every $j\in{\mathbb{N}}$, so that $\textup{Osc}^{\lambda}_{\mu_{1}^{l}}(\bar{w},R)=0$

Note moreover that (6.2) can be extended to every ball $B_{R}(p)$ with $p\in B_{22t}(w_{0})$ and $R\in(0,22t]$: indeed, if $B_{R}(p)\cap{\rm spt}\,(\mu_{1}^{l})=\emptyset$, then $\textup{Osc}^{\lambda}_{\mu_{1}^{l}}(p,R)=0$; otherwise, if $w\in B_{R}(p)\cap\sup(\mu_{1}^{l})$, then $B_{R}(p)\subset B_{2R}(w)$ and

[TABLE]

being $\beta(y,\rho)\leq 2^{n+1}\beta(y,2\rho)$ for every $y$ and every $\rho>0$.

The conclusion of the proof is now an application of the following result by Naber–Valtorta [31, Theorem 3.4 & Remark 3.9].

Theorem 6.3** (Naber–Valtorta [31]).**

There is a dimensional constant $C_{\ref{t:Reif discreto}}(n)>0$ such that the following holds. For every $\lambda>0$, there exists $\delta_{\ref{t:Reif discreto}}(\lambda)>0$ with this property: for every $\{B_{r_{i}}(x_{i})\}_{i\in I}$ finite collection of pairwise disjoint balls in $B_{2}\subset\mathbb{R}^{n}$ and $\mu:=\sum_{i\in I}r_{i}^{n-1}\delta_{x_{i}}$,

[TABLE]

Renaming for simplicity the points in the support of $\mu_{1}^{l}$ as $\mu^{l}_{1}=\sum_{i}s_{p_{i}}^{n-1}\,\delta_{p_{i}}$, we can apply Theorem 6.3 with $x_{i}:=\nicefrac{{(p_{i}-w_{0})}}{{(11t)}}$, $r_{i}:=\nicefrac{{s_{p_{i}}}}{{(440t)}}$, and $\mu:=\sum_{i}r_{i}^{n-1}\delta_{x_{i}}$. Indeed, from (6.15) we have that ${\rm spt}\,(\mu)\subset B_{1}$ and from (6.14) we have that $B_{r_{i}}(x_{i})$ are pairwise disjoint. Moreover, from (6.26) and the choice of $\tau$ it follows that, for every $B_{r}(x)\subset B_{2}$ we have

[TABLE]

We then conclude that $\mu^{l}_{1}(B_{11t}(w_{0}))=(440t)^{n-1}\mu(B_{1})\leq C_{\ref{t:Reif discreto}}\,(440t)^{n-1}$ and, by the choice of the constant $C_{\ref{l:misura 2}}$, we conclude (6.10):

[TABLE]

7. Structure of the free boundary ${\mathcal{H}}^{n-1}$-a.e.

In this section we give the proof of Theorem 1.2. It is a consequence of Theorem 1.1 and of the following rectifiability criterion recently established by Azzam–Tolsa [6, Theorem 1.1]. A similar criterion has also been established independently by Naber–Valtorta in [31, Theorem 3.3].

7.1. Azzam–Tolsa rectifiability criterion

We recall the following definition: a Radon measure $\mu$ in $\mathbb{R}^{n}$ is called $k$-rectifiable if

• (i)

$\mu$ is absolutely continuous with respect to the Hausdorff measure ${\mathcal{H}}^{k}$, i.e. for every $E\subset\mathbb{R}^{n}$

[TABLE]

• (ii)

there exist at most countable many $C^{1}$ functions $f_{i}:\mathbb{R}^{k}\to\mathbb{R}^{n}$, for $i\in{\mathbb{N}}$, such that

[TABLE]

A set $E\subset\mathbb{R}^{n}$ is said ${\mathcal{H}}^{k}$-rectifiable if the associated measure ${\mathcal{H}}^{k}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits E$ is $k$-rectifiable.

The following is the rectifiability criterion we are going to exploit: in order to state it, we need to recall the notion of upper-density of a measure

[TABLE]

Theorem 7.1** (Azzam–Tolsa [6]).**

Let $\mu$ be a finite Borel measure in $\mathbb{R}^{n}$ with $\vartheta^{k,\star}(x,\mu)<+\infty$ for $\mu$-a.e. $x\in\mathbb{R}^{n}$. Then, $\mu$ is $k$-rectifiable if

[TABLE]

The following two remarks are in order.

Remark 7.2*.*

In the case $E\subset\mathbb{R}^{n}$ is a Borel set with ${\mathcal{H}}^{k}(E)<+\infty$, then $\mu:={\mathcal{H}}^{k}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits E$ has upper-density finite $\mu$-almost everywhere. More precisely, $\vartheta^{k,\star}(x,\mu)\leq 1$ for $\mu$-a.e. $x\in E$ (see for instance [2, (2.43)]).

Remark 7.3*.*

Let $\lambda\in(0,1)$ be any number. For every $\lambda^{q+1}\leq r<\lambda^{q}$ (with $q\in{\mathbb{N}}$) we have that $\beta_{\mu,2}^{(k)}(x,r)\leq C\,\beta_{\mu,2}^{(k)}(x,\lambda^{q})$ for some constant $C=C(\lambda,k)$, and hence

[TABLE]

We can now prove that $\Gamma(u)$ is ${\mathcal{H}}^{n-1}$-rectifiable.

7.2. Proof of Theorem 1.2

We are given a solution $u$ to the lower dimensional obstacle problem in $B_{1}$ and we want to show that $\Gamma(u)\cap B_{R}$ is rectifiable for every $R<1$. Set $\delta:=\nicefrac{{(1-R)}}{{2}}$ and let $\{B_{\delta}(x_{i})\}_{i\in J}$ be a finite covering of $\Gamma(u)\cap B_{R}$, with $x_{i}\in\Gamma(u)$, and set $L:=\max_{i\in J}\Theta_{u}(x_{i},2\delta)$. Then, it suffices to show that $\Gamma(u)\cap B_{\delta}(x_{i})$ is rectifiable.

After a suitable change of variable ($v(x):=u(x_{i}+\delta\,x)$ – cf. Remark 2.6), we are left to verify the following statement: let $v$ be a solution to the lower dimensional obstacle problem in $B_{2}$ with $0\in\Gamma(v)$, then $\Gamma(v)\cap B_{1}$ is rectifiable. To this aim, for every $l\in{\mathbb{N}}\setminus\{0\}$ we consider the following sets:

[TABLE]

Note that $E_{l}\subseteq E_{l+1}$; and that Theorem 1.1 and Remark 7.2 imply

[TABLE]

Therefore, it is now enough to show that $E_{l}$ is ${\mathcal{H}}^{n-1}$-rectifiable for any fixed integer $l\in{\mathbb{N}}$; in this respect we set $\mu_{l}:={\mathcal{H}}^{n-1}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits E_{l}$. We fix $\lambda\in(0,\nicefrac{{1}}{{18}})$ and an integer $q_{0}$ such that $\lambda^{q_{0}-1}\leq 1$. By applying Proposition 4.2 (with parameter $R=7$) we have that

[TABLE]

where we used: Fubini’s Theorem and $1+\lambda^{q_{0}}\leq\nicefrac{{3}}{{2}}$ in the second inequality, $18\lambda<1$ in the third, and $\lambda^{q_{0}-1}\leq 1$ and $\mu(B_{\nicefrac{{3}}{{2}}})<+\infty$ (by Theorem 1.1) in the last line. The conclusion now follows straightforwardly: indeed, by (7.2) we have that

[TABLE]

In view of (7.2), we can then apply Theorem 7.1 to conclude that $E_{l}$ is ${\mathcal{H}}^{n-1}$-rectifiable. ∎

Remark 7.4*.*

The rectifiability of the free boundary can also be deduced by following the argument of Naber–Valtorta [31, 32], along the proof of the covering argument and the discrete Reifenberg Theorem: we refer to [31, 32] for more details.

8. Classification of blow-ups ${\mathcal{H}}^{n-1}$-a.e.

In this section we give the proof of the last main result of the paper, namely Theorem 1.3. We recall the rescalings for the blow-up procedure:

[TABLE]

In view of Remark 2.14, the functions $\bar{u}_{x_{0},r}$ and $u_{x_{0},r}$ have limits which differ only by a multiplicative constant. Therefore, Theorem 1.3 is proven once we show the same conclusions for the new rescalings (8.1).

8.1. Stratification of the free boundary

We start off with the first part of Theorem 1.3 regarding the estimate on the dimension of the set of points with frequency $\lambda\in\{2m,2m-1+s,2m+2s\}_{m\in{\mathbb{N}}\setminus\{0\}}$. We use a stratification argument for the nodal set $\mathcal{N}(u)$ of a solution $u$ to the lower dimensional obstacle problem (1.1). This argument goes back to the work of Almgren [1, § 2.26]; here for convenience we follow [19].

We start recalling the definition of nodal points:

[TABLE]

Next we specify the main ingredients of [19, § 3.1] for the thin obstacle problem:

• (a)

the upper semi-continuous function $f:B_{1}^{\prime}\to\mathbb{R}$ given by

[TABLE]

• (b)

the compact class of conical functions $\mathcal{G}(x)\subset L^{\infty}(\mathbb{R}^{n})$, for every $x\in B_{1}^{\prime}$, defined by

[TABLE]

recall that $\text{BU}(x)$ denotes the set of all blow-ups of $u$ at $x$.

We need to verify that ${\mathcal{G}}(x)$ is a class of compact conical functions according to [19, Definition 3.3] (the arguments are analogous to those in [19, § 5.2], we repeat them for readers’ convenience).

• (1)

An upper semi-continuous function $g:\mathbb{R}^{n}\to\mathbb{R}$ is said to be conical if $g(z)=g(0)$ implies that

[TABLE]

Then, both the zero function and the frequency of homogeneous solutions $w$ are conical by Lemma 5.2.

• (2)

A class ${\mathcal{G}}$ of conical functions is called compact if for every sequence $(g_{j})_{j\in{\mathbb{N}}}\subset{\mathcal{G}}$ there exist a subsequence $(g_{j_{i}})_{i\in{\mathbb{N}}}\subset(g_{j})_{j\in{\mathbb{N}}}$ and $g\in{\mathcal{G}}$ such that

[TABLE]

According to item (b), if $(g_{j})_{j\in{\mathbb{N}}}\subset{\mathcal{G}}(x)$ we may assume without loss of generality $g_{j}$ not identically [math] for $j$ big. Then, $g_{j}=I_{w_{j}}(\cdot,0^{+})$ and by Lemma 3.1 and Corollary 2.10 there exists a subsequence $w_{j_{i}}$ converging to a homogeneous solution $w$ (recall that $I_{w_{j}}(1)=I_{u}(x,0^{+})$ and $H_{w_{j}}(1)=1$). By a diagonal argument we have that $w\in\text{BU}(x)$, and (8.2) follows from

[TABLE]

We discuss next the structural hypotheses [19, (i) - (ii) § 3.1]:

• (i)

$g(0)=f(x)$ for all $g\in{\mathcal{G}}(x)$, because $I_{w}(0^{+})=I_{u}(x,0^{+})$ for every blowup $w\in\text{BU}(x)$;

• (ii)

for all $r_{j}\downarrow 0$ there exists a subsequence $(r_{j_{i}})_{i\in{\mathbb{N}}}\subset(r_{j})_{j\in{\mathbb{N}}}$ and $w\in{\mathcal{G}}(x)$ such that $u_{x,r_{j_{i}}}\to w$; hence, for every $y\in\mathbb{R}^{n}$ and for every sequence $y_{i}\to y$, we have

[TABLE]

We are then in the position to apply [19, Theorem 3.4] and conclude that the points whose blow-ups have spines with dimension not exceeding $l\in\{0,\ldots,n\}$ constitute a set of Hausdorff dimension at most $l$.

Theorem 8.1**.**

Let $u$ be a solution of the thin obstacle problem (1.1) in $B_{R}$. For $l\in\{0,\ldots,n\}$, set $\Sigma_{l}(u):=\{x\in\mathcal{N}(u):\,\dim S(w)\leq l,\;\forall\;w\in\textup{BU}(x)\}$. Then, $\Sigma_{0}(u)$ is at most countable and $\dim_{{\mathcal{H}}}\Sigma_{l}(u)\leq l$ for every $l\in\{1,\ldots,n\}$.

The first assertion of Theorem 1.3 is now a direct consequence.

Proof of Theorem 1.3: part I.

We first show that $\dim S(w)\leq n-1$ for every $w\in\textup{BU}(x)$ with $x\in\mathcal{N}(u)$. To this aim, we observe that by the definition of nodal set we have that $0\in\mathcal{N}(w)$ for every $w\in\textup{BU}(x)$ with $x\in\mathcal{N}(u)$. On the other hand, using the notation in Theorem 8.1, $\Sigma_{n}(u)\setminus\Sigma_{n-1}(u)=\emptyset$ as noticed in Section 5.2. Indeed, the only non-trivial homogeneous solutions with $n$-dimensional spine are the functions $w_{c}:=c\,|x_{n+1}|^{2s}$ with $c<0$, and by direct computation $\mathcal{N}(w_{c})=\emptyset$.

Therefore, for every $x\in\Gamma(u)\setminus\Sigma_{n-2}(u)$ there exists at least a blowup $w\in\textup{BU}(x)$ with an $(n-1)$-dimensional spine $S(w)$, i.e. with $w\in{\mathcal{H}}^{\textup{top}}$. Thus, by the classification of all homogeneous solutions with maximal spine in Lemma 5.3, the limiting frequency at any point $x\in\Gamma(u)\setminus\Sigma_{n-2}(u)$ satisfies

[TABLE]

Taking into consideration that $\dim_{{\mathcal{H}}}\Sigma_{n-2}(u)\leq n-2$ by Theorem 8.1, we conclude the proof. ∎

8.2. Uniqueness of blow-ups with frequency $2m$ and $2m-1+s$

For the second part of Theorem 1.3 we need an extension of the classification result in Lemma 5.3 to the $\lambda$-homogeneous (even symmetric with respect to $x_{n+1}$) solutions of

[TABLE]

with $\lambda\in\{2m,2m-1+s\}_{m\in{\mathbb{N}}\setminus\{0\}}$ and $\{x\cdot e=x_{n+1}=0\}\subseteq\Lambda(u)$ for some unit vector $e\in\mathbb{R}^{n}\times\{0\}$. The main differences with Lemma 5.3 are that neither the unilateral obstacle condition nor any invariance assumption of the solutions (i.e. the assignment of the spine) are imposed in this framework. In the ensuing statement we keep the notation introduced in Lemma 5.3.

Proposition 8.2**.**

Let $u:\mathbb{R}^{n+1}\to\mathbb{R}$ be a non-trivial $\lambda$-homogeneous weak solution of (8.3), even w.r.to $x_{n+1}$, such that $\lambda\in\{2m,2m-1+s\}_{m\in{\mathbb{N}}\setminus\{0\}}$ and $\{x\cdot e=x_{n+1}=0\}\subseteq\Lambda(u)$ for some unit vector $e\in\mathbb{R}^{n}\times\{0\}$. Then, there exists $c>0$ such that $u(x)=c\,h_{2m}(x\cdot e,x_{n+1})$ or $u(x)=c\,h_{2m-1+s}(x\cdot e,x_{n+1})$ or $u(x)=c\,h_{2m-1+s}(-x\cdot e,x_{n+1})$.

The proof is postponed to Proposition A.3 in the appendix. Given it for granted, we proceed with the conclusion of the proof of Theorem 1.3.

Proof of Theorem 1.3: part II.

By Theorem 1.2 there exist at most countably many $C^{1}$-regular submanifolds $\{M_{i}\}_{i\in{\mathbb{N}}}$ such that ${\mathcal{H}}^{n-1}(\Gamma(u)\setminus\cup_{i\in{\mathbb{N}}}M_{i})=0$. We consider the sets $\Gamma_{i}(u):=\Gamma(u)\cap M_{i}$ and

[TABLE]

Note that ${\mathcal{H}}^{n-1}(\Gamma_{i}(u)\setminus\Gamma_{i}^{\prime}(u))=0$ for every $i\in{\mathbb{N}}$ by Besicovitch’s differentiation theorem (cp. [2, Theorem 2.22]). We show that for every $i\in{\mathbb{N}}$ and for every $x_{0}\in\Gamma_{i}^{\prime}(u)$ the conclusion of Theorem 1.3 holds, namely if $I(x_{0},0^{+})=\lambda\in\{2m,2m-1+s\}_{m\in{\mathbb{N}}\setminus\{0\}}$, then there exists a unit vector $e_{x_{0}}$ with $e_{x_{0}}\perp\text{Tan}_{x_{0}}M_{i}$ at $x_{0}$ such that

[TABLE]

where $h_{\lambda}$ are the functions in Lemma 5.3, and $\text{Tan}_{x_{0}}M_{i}$ is the linear tangent space to $M_{i}$ at $x_{0}$: i.e.

[TABLE]

To this aim we consider the compact sets

[TABLE]

By Blaschke compactness theorem (cp. [2, Theorem 6.1]) the sequence of sets $\{K_{r}\}_{r\in(0,r_{0}]}$ is pre-compact in the Hausdorff distance on $\overline{B}_{1}$: namely, given any sequence $(0,r_{0}]\ni r_{i}\to 0$, there exists a subsequence $(r_{i_{k}})_{k\in{\mathbb{N}}}$ and a compact set $K_{0}\subseteq\overline{B}_{1}$ such that we have $\lim_{k}{\textup{dist}}_{{\mathcal{H}}}(K_{r_{i_{k}}},K_{0})=0$, i.e.

• (a)

any point $x\in K_{0}$ is an accumulation point for a sequence $(x_{k})_{k\in{\mathbb{N}}}$ with $x_{k}\in K_{r_{i_{k}}}$;

• (b)

if $x_{k}\in K_{r_{i_{k}}}$, then any accumulation point of $(x_{k})_{k\in{\mathbb{N}}}$ belongs to $K_{0}$.

We proceed now with the proof of (8.4) in three steps.

1. Let $r_{j}\downarrow 0$ be such that ${\textup{dist}}_{{\mathcal{H}}}(K_{r_{j}},K_{0})\to 0$ for some compact set $K_{0}$. Then

[TABLE]

Assuming this is not the case, there exists an open ball $B_{\rho}(y_{0})\subset B_{1}\subset\mathbb{R}^{n}$ with $y_{0}\in\text{Tan}_{x_{0}}M_{i}$ such that \big{(}\Gamma_{i}(u)-x_{0}\big{)}/r_{j}\cap B_{\rho}(y_{0})=\emptyset. In particular, for sufficiently large $j$ we have that

[TABLE]

since {\mathcal{H}}^{n-1}\big{(}\nicefrac{{\big{(}M_{i}-x_{0}\big{)}}}{{r_{j}}}\cap A\big{)}\to{\mathcal{H}}^{n-1}\big{(}\text{Tan}_{x_{0}}M_{i}\cap A\big{)} for every open set $A\subset\mathbb{R}^{n}$. This yields

[TABLE]

against the assumption $x_{0}\in\Gamma^{\prime}_{i}(u)$.

2. In particular, it follows that, if $u_{x_{0}}$ is any blow-up limit of $u$ at $x_{0}\in\Gamma^{\prime}_{i}(u)$, then

[TABLE]

Indeed, set Y_{r}:=\big{\{}u_{x_{0},r}=0\big{\}}\cap\overline{B}_{1} and note that $K_{r}\subset Y_{r}$. If $Y_{0}\subset\overline{B}_{1}$ is any Hausdorff limit of a sequence $Y_{r_{j}}$, then Y_{0}\subset\big{\{}u_{x_{0}}=0\big{\}}\cap\overline{B}_{1}, because $u_{x_{0},r_{j}}(z_{j})\to u_{x_{0}}(z_{0})$ for every $z_{j}\in Y_{r_{j}}$ with $z_{j}\to z_{0}$ (thanks to the uniform convergence of $u_{x_{0},r_{i_{k}}}$). In particular, being $K_{r_{j}}\subset Y_{r_{j}}$, the conclusion follows from step 1 and the homogeneity of $u_{x_{0}}$.

3. We now conclude the proof of (8.4). Assume without loss of generality that $\text{Tan}_{x_{0}}M_{i}=\{x_{n}=x_{n+1}=0\}$. By Proposition 2.11 we have that $\textup{BU}(x_{0})\subseteq{\mathcal{H}}_{\lambda}$ with $\lambda=I_{u}(x_{0},0^{+})$, and we distinguish two possibilities (recall also that the blow-ups are renormalized so to have $H_{u_{x_{0}}}(1)=1$):

• (1)

$I_{u}(x_{0},0^{+})=2m$. By Proposition 8.2 the blow-up $u_{x_{0}}$ needs to be $h_{2m}(x_{n},x_{n+1})$, because this function is the only blow-up with frequency $2m$ and contact set containing $\text{Tan}_{x_{0}}M_{i}=\{x_{n}=x_{n+1}=0\}$ by (8.5);

• (2)

$I_{u}(x_{0},0^{+})=2m-1+2s$. By Proposition 8.2 every blow-up $u_{x_{0}}$ is given by either $h^{+}=h_{2m-1+s}(x_{n},x_{n+1})$ or $h^{-}=h_{2m-1+s}(-x_{n},x_{n+1})$.

In order to infer the uniqueness of the blowup in this last case, we exploit the connectedness of the set of blow-up limits. Namely, assume that there exist $r_{i}\downarrow 0$ and $\rho_{i}\downarrow 0$ such that $u_{x_{0},r_{i}}\to h^{+}$ and $u_{x_{0},\rho_{i}}\to h^{-}$; up to passing to subsequences, we may take $r_{i}<\rho_{i}<r_{i+1}$. Then, by continuity there exists $t_{i}\in(r_{i},\rho_{i})$ such that

[TABLE]

Since the sequence $(u_{x_{0},t_{i}})_{i\in{\mathbb{N}}}$ has no subsequence converging either to $h^{+}$ or to $h^{-}$, this gives a contradiction and concludes the proof of Theorem 1.3. ∎

8.3. Concerning the optimality of Theorem 1.3

For every $e\in\mathbb{R}^{n+1}$ with $|e|=1$ and $e\cdot e_{n+1}=0$, the functions $u(x)=h_{2m}(x\cdot e,x_{n+1})$ and $u=h_{2m-1+s}(x\cdot e,x_{n+1})$ are examples of solutions to the lower dimensional problem (1.1) in any ball $B_{R}$ whose free boundary $\Gamma(u)$ is $(n-1)$-dimensional and is made of points with frequency $2m$ and $2m-1+s$, respectively. Note that the latters are explicit cases in which $\Gamma(u)=\textup{Other}(u)$.

On the other hand, as pointed out in the introduction, at the best of our knowledge there are no explicit examples of solutions to the lower dimensional obstacle problem (1.1) with free boundary points with frequency $2m+2s$ with $m\in{\mathbb{N}}\setminus\{0\}$ (note that, although $h_{2m+2s}(x\cdot e,x_{n+1})$ are solutions, $\Gamma(h_{2m+2s})=\emptyset$).

Such points do not occur in the one dimensional case $n=1$. Following the argument of [23, Remark 1.2.8] for $s=\nicefrac{{1}}{{2}}$, assume that $0\in\Gamma(u)$ is a point with frequency $2m+2s$. Then, one can find a sequence $(t_{k},0)\in\mathbb{R}^{2}$ with $\lim_{k}t_{k}=0$ such that $u(t_{k},0)>0$ and, therefore, from (2.4),

[TABLE]

Taking the rescalings $u_{0,\nicefrac{{t_{k}}}{{2}}}$, up to passing to a subsequence (not relabeled) there exists a blowup $w\in\textup{BU}(0)$ such that (cp. (2.20)):

[TABLE]

Note that necessarily $w=h_{2m+2s}$, because there exists a unique blowup with frequency $2m+2s$. Moreover, from (8.6) we have that $\lim_{x_{2}\downarrow 0}x_{2}^{a}\partial_{x_{2}}w(\nicefrac{{1}}{{2}},x_{2})=0$. On the contrary a direct computation shows that $\lim_{x_{2}\downarrow 0}x_{2}^{a}\partial_{x_{2}}h_{2m+2s}(\nicefrac{{1}}{{2}},x_{2})<0$, thus leading to a contradiction and implying that there cannot exist free boundary points with frequency $2m+2s$ for $n=1$.

Potential points with frequency $2m+2s$ are sometimes referred to in the literature as degenerate points (see the final section of [23]). It is a tempting conjecture to claim that there are actually none. If this were the case, Theorem 1.3 would then be optimal, both concerning the uniqueness of blow-ups at ${\mathcal{H}}^{n-1}$-almost all points of the free boundary, and the classification of the frequency at ${\mathcal{H}}^{n-2}$-almost all points of the free boundary.

Appendix A Homogeneous solutions

In this appendix we collect some results concerning homogeneous solutions to the thin obstacle problem and more generally to the corresponding system of Euler–Lagrange equations. Therefore, we consider functions $u:\mathbb{R}^{n+1}\to\mathbb{R}$ such that

[TABLE]

for some $\lambda\geq 1+s$ (this restriction being in accordance with the homogeneity of all possible blow-ups).

A.1. Two-dimensional homogeneous solutions

Here we provide a classification of the homogeneous solutions to the equation

[TABLE]

in the two dimensional case, i.e. for $n=1$. Thus, necessarily, the contact set is a cone, and we have:

• (i)

$\Lambda(u)=\{x_{1}=x_{2}=0\}$,

• (ii)

$\Lambda(u)=\{x_{1}\leq 0,\,x_{2}=0\}$ or $\Lambda(u)=\{x_{1}\geq 0,\,x_{2}=0\}$,

• (iii)

$\Lambda(u)=\{x_{2}=0\}$.

Correspondingly, we introduce three classes of functions $\Phi_{m}$, $\Psi_{m}$ and $\Pi_{m}$ for $m\in{\mathbb{N}}$, that are explicitly defined as follows:

[TABLE]

where $\lfloor\cdot\rfloor$ denotes the integer part of a real number, and

[TABLE]

[TABLE]

[TABLE]

and the (increasing) Pochhammer symbol is defined by

[TABLE]

We establish the ensuing classification result (for related issues see [12, 34]).

Proposition A.1**.**

Let $u:\mathbb{R}^{2}\to\mathbb{R}$ be $\lambda$-homogeneous, even symmetric w.r.to $x_{2}$, and assume that $u$ is a weak solution of (A.1). Then, one of the following occurs:

• (i)

$\Lambda(u)=\{x_{1}=x_{2}=0\}$, $\lambda=m\in{\mathbb{N}}\setminus\{0\}$ and $u$ is a multiple of $\Phi_{m}$;

• (ii)

$\Lambda(u)=\{x_{1}\leq 0,\,x_{2}=0\}$* (resp. $\Lambda(u)=\{x_{1}\geq 0,\,x_{2}=0\}$), $\lambda=m+s$ for some $m\in{\mathbb{N}}$ and $u$ is a multiple of $\Psi_{m}$ (resp. of $\Psi_{m}(-x_{1},x_{2})$);*

• (iii)

$\Lambda(u)=\{x_{2}=0\}$, $\lambda=m+2s$ for some $m\in{\mathbb{N}}$ and $u$ is a multiple of $\Pi_{m}$.

Moreover, if $u$ is a solution to the lower dimensional obstacle problem, then $m$ is even in (i) and (iii), and $m$ is odd in (ii).

For the proof we need to introduce the hypergeometric function ${}_{2}F_{1}(\alpha,\beta;\gamma;\cdot):{\mathbb{C}}\to{\mathbb{C}}$ defined by

[TABLE]

where $\alpha,\beta,\gamma\in{\mathbb{C}}$, $\gamma$ not a negative integer. The power series defining ${}_{2}F_{1}$ is converging for $|z|<1$, and it can be analytically continued elsewhere. In what follows we shall use several properties of ${}_{2}F_{1}$ for which we refer to the Digital Library of Mathematical Functions, always quoting the precise formulas employed in the derivation and referring to their enumeration in [33].

We warn the reader that, with a slight abuse of notation, in this section $\Gamma$ shall denote both the free boundary of a solution and the Euler’s Gamma-function on the complex plane, extended to $Re(z)\leq 0$ by analytic continuation using the identity $\Gamma(1+z)=z\Gamma(z)$. In particular, $\Gamma$ turns out to be a meromorphic function with no zeros and simple poles at $z=-m$, $m\in{\mathbb{N}}$. Thus, we adopt the convention that $\Gamma^{-1}(-m)=0$ for all $m\in{\mathbb{N}}$.

Proof of Proposition A.1.

Using polar coordinates $x_{1}=r\cos\theta$ and $x_{2}=r\sin\theta$ with $r>0$ and $\theta\in[0,\pi]$, let v(r,\theta):=u\big{(}r\cos\theta,r\sin\theta\big{)}=r^{\lambda}\,y(\theta). The Euler-Lagrange equation (A.1) then reads as

[TABLE]

with boundary conditions:

• •

case (i)

[TABLE]

• •

case (ii) (by symmetry we assume $\Lambda(u)=\{x_{1}\leq 0,x_{2}=0\}$)

[TABLE]

• •

case (iii)

[TABLE]

The change of variable $y(\theta)=(\sin\theta)^{s}\,h(\cos\theta)$ transforms the ODE for $y$ in (A.6) into an associated Legendre differential equation for $h$. More precisely, we get for $\nu=\lambda-s$ and $\nu\geq 1$

[TABLE]

with the following boundary conditions:

• •

case (i)

[TABLE]

• •

case (ii)

[TABLE]

• •

case (iii)

[TABLE]

The associated Legendre equation can be solved explicitly in terms of the hypergeometric function ${}_{2}F_{1}$ (cf. (A.5)). A generic solution in the interval $(-1,1)$ is given by

[TABLE]

where $A_{1},\,A_{2}\in\mathbb{R}$ and

[TABLE]

(cf. [33, (14.3.1), (14.3.2), Section 15.1 and (15.8.1)]).

1. Dirichlet boundary conditions. We now proceed computing the boundary conditions in terms of the explicit representations (A.14) and (A.15). First, note that by continuity of ${}_{2}F_{1}(\alpha,\beta,\gamma,\cdot)$ and since ${}_{2}F_{1}(\alpha,\beta,\gamma,0)=1$ for all $\alpha$, $\beta$ and $\gamma$, we get

[TABLE]

from which we get

[TABLE]

For the corresponding limit values as $x\downarrow-1$ we use [33, (15.4.20)] to infer

[TABLE]

and from (A.15) and [33, (15.4.20)]

[TABLE]

from which

[TABLE]

2. Neumann boundary conditions. For what concerns the boundary conditions involving the derivative of $h$ we use [33, (15.5.1)] to compute

[TABLE]

Hence, we get

[TABLE]

From the latter formula we immediately conclude that

[TABLE]

and

[TABLE]

Therefore, we have

[TABLE]

In addition, from (A.1), from the linear transformation of variable rule for ${}_{2}F_{1}$ in [33, (15.8.4)] and from [33, Section 15.5], elementary calculations lead to

[TABLE]

In turn, this implies

[TABLE]

Finally, by [33, (15.8.1)] we rewrite (A.1) for $P^{-s}_{\nu}$ as

[TABLE]

and infer from [33, (15.4.20)]

[TABLE]

i.e.

[TABLE]

3. By means of (A.16), (A.17), (A.19) and (A.1) we are able to complete the classification by discussing all the cases (i) – (iii). We start off with case (i): using (A.19) and (A.1) we deduce that $A_{2}=0$ and $\nu+s=m\in{\mathbb{N}}$ (in order to have $\nicefrac{{1}}{{\Gamma(-s-\nu)}}=0$). Therefore,

[TABLE]

In particular, $(1-x^{2})^{\nicefrac{{s}}{{2}}}h(x)$ is a polynomial of degree $d\leq m$ (or a constant if $m=0$), as $(-m)_{k}=0$ for every $k\geq m+1$. The case $m=0$ implies $y$ to be constant and thus $u\equiv 0$, which is excluded from the condition $\Lambda(u)=\{x_{1}=x_{2}=0\}$. Hence, $m>0$ and $y$ is a polynomial of degree $d$ in $\cos\theta$. As for every $k\geq 0$

[TABLE]

we infer that $d=\lambda=\nu+s=m$ and that $y$ depends only on powers of $\cos\theta$ with the same parity as $m$:

[TABLE]

Therefore, $u$ is an $m$-homogeneous polynomial of the form in (A.2) and by a direct computation

[TABLE]

we conclude the explicit form of the coefficients $\alpha_{k}$.

Next we discuss case (ii): from (A.19) we get $A_{2}=0$ and from (A.17) we get $\nu\in{\mathbb{N}}$. Thus ${}_{2}F_{1}(\nu+1,-\nu,1-s,\cdot)$ is a polynomial of degree at most $\nu=m$ with $m\in{\mathbb{N}}\setminus\{0\}$. The corresponding representation formula in (A.3) follows at once from

[TABLE]

We discuss case (iii): from (A.16) and (A.17) we get that $A_{1}=0$ and $\nu-s=m\in{\mathbb{N}}$ and the representation formula for solutions in (A.4) follows by direct verification (alternatively one can also derive it from the explicit formula in terms of the hypergeometric function).

4. Finally, we discuss the case of solutions $u$ to the lower dimensional obstacle problem (1.1). In particular, $u$ solves (A.1), $u|_{B_{1}^{\prime}}\geq 0$ and the normal weighted derivative satisfies a sign condition. Thus, the following additional boundary conditions need to be satisfied by $y$:

[TABLE]

In turn, these for the function $h$ translate into

[TABLE]

We can then discuss the implications of (A.21) – (A.24) for the tree cases (i) – (iii). In case (i), by (A.16) and (A.23) we get $A_{1}\geq 0$; similarly, by (A.17) and (A.24) we get that $\Gamma(-\nu)>0$, i.e. $2m-1\leq\nu\leq 2m$ for some $m\in{\mathbb{N}}\setminus\{0\}$. Since $\nu+s\in{\mathbb{N}}$, we conclude that $\nu+s=2m$.

In case (ii), using (A.16) and (A.23) we conclude that $A_{1}\geq 0$; moreover, from (A.1) and (A.22) we infer that $\Gamma(-s-\nu)\geq 0$ and therefore $2m+1\leq\nu+s\leq 2m+2$ for some $m\in{\mathbb{N}}$. In particular, since $\nu\in{\mathbb{N}}$, we have that $\nu=2m+1$ is odd.

Finally, in case (iii), using (A.19) in (A.21) and (A.1) in (A.22) we deduce that $A_{2}\geq 0$ and $\Gamma(1-\nu)\leq 0$, from which it follows that $\nu-s=2m$ is even. ∎

Using Proposition A.1 we now complete the proof of the classification of global solutions $u\in{\mathcal{H}}^{top}$ with $(n-1)$-dimensional spine in Lemma 5.3.

Proof of Lemma 5.3.

For every $u\in{\mathcal{H}}^{top}$, we have that $u$ depends on $x_{n+1}$ and only one in-plane variable, i.e. $u(x)=v(x\cdot e,x_{n+1})$ for some $v:\mathbb{R}^{2}\to\mathbb{R}$ and for some unit vector $e\in\mathbb{R}^{n+1}$ with $e\cdot e_{n+1}=0$. In particular, $v$ is a two-dimensional solution to the lower dimensional obstacle problem in $\mathbb{R}^{2}$. Therefore, by Proposition A.1 we know that $\lambda\in\{2m,2m-1+s,2m+2s\}_{m\in{\mathbb{N}}\setminus\{0\}}$ and $v$ is one of the functions in (A.2) – (A.4). The statements about $\Gamma(u)$, $\Lambda(u)$, $\mathcal{N}(u)$ and $S(u)$ follow from the explicit formulas therein. ∎

A.2. Further classification results

Here we provide a proof to Proposition 8.2. We split the argument in two parts. We start off classifying in any dimension all $\lambda$-homogeneous solutions (even symmetric with respect to $x_{n+1}$) of

[TABLE]

such that $\lambda\in[1+s,2+s)$ and having as contact set $\Lambda(u)$ one of the following

• (i)

$\Lambda(u)=\{x_{n}=x_{n+1}=0\}$,

• (ii)

$\Lambda(u)=\{x_{n}\leq 0,\,x_{n+1}=0\}$,

• (iii)

$\Lambda(u)=\{x_{n+1}=0\}$.

We follow the arguments in [12, Lemma 5.3] and [24, Lemma A.3], in which the case $\lambda=1+s$ with $\Lambda(u)=\{x_{n}\leq 0,\,x_{n+1}=0\}$ is addressed. To this aim we introduce the following notation: $x=(x^{\prime\prime},x_{n},x_{n+1})\in\mathbb{R}^{n-1}\times\mathbb{R}\times\mathbb{R}$.

Lemma A.2**.**

Let $u:\mathbb{R}^{n+1}\to\mathbb{R}$ be a $\lambda$-homogeneous solution of (A.25), even symmetric w.r.to $x_{n+1}$, with $\lambda\in[1+s,2+s)$ and $\Lambda(u)$ one of the sets in (i) – (iii) above. Then, the following occurs:

• •

in case (i), $\lambda=2$ and there exists a $1$-homogeneous polynomial $q:\mathbb{R}^{n-1}\to\mathbb{R}$ and a constant $c\in\mathbb{R}$ such that

[TABLE]

• •

in case (ii), $\lambda=1+s$ and there exists a $1$-homogeneous polynomial $q:\mathbb{R}^{n-1}\to\mathbb{R}$ and a constant $c\in\mathbb{R}$ such that

[TABLE]

• •

in case (iii), $\lambda=1+2s$ and there exists a $1$-homogeneous polynomial $q:\mathbb{R}^{n}\to\mathbb{R}$ such that

[TABLE]

Proof.

1. In case (i), since ${\mathcal{H}}^{n}(\Lambda(u))=0$, it follows from [12, Lemma 5.3] that $u$ is a polynomial. Therefore, $\lambda=2$ and by symmetry $u(x)=q(x^{\prime},x_{n})+\alpha\,x_{n+1}^{2}$, with $q:\mathbb{R}^{n}\to\mathbb{R}$ a $2$-homogeneous polynomial and $\alpha\in\mathbb{R}$. Furthermore, by taking into account that $\Lambda(u)=\{x_{n}=x_{n+1}=0\}$ we infer that $q(x^{\prime},x_{n})=q_{1}(x^{\prime})x_{n}+\beta\,x_{n}^{2}$, with $q_{1}:\mathbb{R}^{n-1}\to\mathbb{R}$ a $1$-homogeneous polynomial and $\beta\in\mathbb{R}$. Thus, $u(x)=q_{1}(x^{\prime})x_{n}+\beta\,x_{n}^{2}+\alpha\,x_{n+1}^{2}$, and imposing that $u$ solves the equation we conclude that

[TABLE]

In particular, from the classification in Proposition A.1, we must have $\beta\,x_{n}^{2}+\alpha\,x_{n+1}^{2}=c\,\Phi_{2}$, thus implying (A.26).

2. In case (ii), we consider the tangential derivatives up to the third order $\partial_{i}u$, $\partial_{ij}u$ and $\partial_{ijk}u$ in directions $i,j,k\in\{1,\ldots,n-1\}$. By the regularity estimate in [17] (cf. also [24, Lemma A.2]) we deduce that $\partial_{i}u$, $\partial_{ij}u$ and $\partial_{ijk}u\in H^{1}_{\textup{loc}}(\mathbb{R}^{n+1},|x_{n+1}|^{a}\,{\mathcal{L}}^{n+1})\cap L^{\infty}_{\textup{loc}}(\mathbb{R}^{n+1})$. In particular, since $\partial_{ijk}u$ is $\lambda-3<0$ homogeneous, it follows that $\partial_{ijk}u\equiv 0$ for all $i,j,k\in\{1,\ldots,n-1\}$. We then infer that

[TABLE]

Being $\partial_{ij}u(0,x_{n},x_{n+1})$ solution to (A.25), the analysis in Proposition A.1 implies that its homogeneity $\lambda-2$ is at least $s$, a condition excluded by the restriction $\lambda<2+s$. We then conclude that $\partial_{ij}u\equiv 0$ for all $i,j,\in\{1,\ldots,n-1\}$, thus we get

[TABLE]

and

[TABLE]

In particular, we infer from Proposition A.1 that the only allowed homogeneity is $\lambda=1+s$, $u(0,x_{n},x_{n+1})=c_{0}\Psi_{1}(x_{n},x_{n+1})$ and $\partial_{i}u(x^{\prime\prime},x_{n},x_{n+1})=c_{i}\Psi_{0}(x_{n},x_{n+1})$, for some constants $c_{i}\in\mathbb{R}$ (note that all these functions solve (A.25) with contact set $\Lambda=\{x_{n}\leq 0,x_{n+1}=0\}$). Using the explicit formulas in Proposition A.1, we conclude (A.27).

3. For case (iii), we can argue analogously as above. In particular, from the $(\lambda-3)$-homogeneity of $\partial_{ijk}u$ and $\lambda-3<0$, it follows that $\partial_{ijk}u\equiv 0$ for all $i,j,k\in\{1,\ldots,n\}$. Therefore, $\partial_{ij}u$ are functions which are $(\lambda-2)$-homogeneous and depend only on $x_{n+1}$. By a direct computation we get from (A.25) that $\partial_{ij}u=c\,x_{n+1}^{2s}$, i.e. $\lambda=2+2s$: since $\lambda<2+s$, we infer that $c=0$ and $\partial_{ij}u=0$ for all $i,j=\{1,\ldots,n\}$, in turn implying

[TABLE]

By taking into account the homogeneity of $u$ and $\partial_{i}u$ and (A.25) (which implies, in particular, that $u(0,x_{n+1})=0$), one then obtains (A.28). ∎

We are now ready to prove the general case of Proposition 8.2. Actually, we show a slightly more general result.

Proposition A.3**.**

Let $u:\mathbb{R}^{n+1}\to\mathbb{R}$ be a non-trivial $\lambda$-homogeneous function even w.r.to $x_{n+1}$. Assume that $u$ is a weak solution of (A.25).

• (i)

If $\lambda=m\in{\mathbb{N}}\setminus\{0,1\}$ and $\{x\cdot e=x_{n+1}=0\}\subseteq\Lambda(u)$ for some unit vector $e\in\mathbb{R}^{n}\times\{0\}$, then

[TABLE]

with $p_{k}:\mathbb{R}^{n-1}\to\mathbb{R}$ harmonic $k$-homogeneous polynomial.

• (ii)

If $\lambda=m+s$, with $m\in{\mathbb{N}}\setminus\{0\}$, and $\{x\cdot e=x_{n+1}=0\}\subseteq\Lambda(u)$ for some unit vector $e\in\mathbb{R}^{n}\times\{0\}$, then

[TABLE]

with $p_{k}:\mathbb{R}^{n-1}\to\mathbb{R}$ harmonic $k$-homogeneous polynomial.

• (iii)

If $\lambda=m+2s$, with $m\in{\mathbb{N}}$, and $\Lambda(u)=\{x_{n+1}=0\}$, then

[TABLE]

with $p:\mathbb{R}^{n}\to\mathbb{R}$ any $m$-homogeneous polynomial and $\gamma_{k}:=\frac{(-1)^{k}}{4^{k}k!\,(1+s)_{k}}$.

Moreover, if $u$ is a solution to the thin obstacle problem (1.1), then in case (i), respectively (ii), $u$ turns out to be a positive multiple of $h_{2m}(x\cdot e,x_{n+1})$, respectively $h_{2m-1+s}(\pm x\cdot e,x_{n+1})$.

Proof.

Without loss of generality we assume that $e=e_{n}$. The proof proceeds by induction on $m\in{\mathbb{N}}$, with starting step provided by Proposition A.2.

The cases (i) and (ii) can be treated by the same argument. We consider the horizontal partial derivatives $\partial_{x_{j}}u$ for $j\in\{1,\ldots,n-1\}$. By the regularity estimate in [17] we have that $\partial_{x_{j}}u\in H^{1}_{\textup{loc}}(\mathbb{R}^{n+1},|x_{n+1}|^{a}\,{\mathcal{L}}^{n+1})\cap L^{\infty}_{\textup{loc}}(\mathbb{R}^{n+1})$ are solutions to (8.3) with $\{x_{n}=x_{n+1}\}\subseteq\Lambda(\partial_{i}u)$ and homogeneity $\lambda-1=m-1$ or $\lambda=m-1+s$, according to the two cases. Using the inductive hypothesis $\partial_{i}u=\sum_{k=0}^{m-1}p_{i,k}\Phi_{m-1-k}$ for $\lambda=m$ or $\partial_{i}u=\sum_{k=0}^{m-1}p_{i,k}^{+}\Psi_{m-1-k}(\cdot,\cdot)+\sum_{k=0}^{m-1}p_{i,k}^{-}\Psi_{m-1-k}(-\cdot,\cdot)$ for $\lambda=m+s$, for some harmonic $k$-homogeneous polynomials $p_{i,k}$ and $p_{i,k}^{\pm}$. Therefore, we infer that

[TABLE]

and similarly

[TABLE]

with $k\,p^{(\pm)}_{k}(x^{\prime\prime}):=\sum_{i=1}^{n-1}p^{(\pm)}_{i,k}(x^{\prime\prime})x_{i}$. Using the equation (A.25) (in particular, recall that $\Phi_{l},\Psi_{l}$ are solutions of (A.25)), we deduce that the polynomials $p_{k},p_{k}^{\pm}$ are harmonic and $u(0,x_{n},x_{n+1})$ is itself a solution (i.e.  $u(0,x_{n},x_{n+1})=c\,\Phi_{m}(x_{n},x_{n+1})$ or $u(0,x_{n},x_{n+1})=c_{1}\,\Psi_{m}(x_{n},x_{n+1})+c_{2}\,\Psi_{m}(-x_{n},x_{n+1})$ for some $c,c_{1},c_{2}\in\mathbb{R}$), thus concluding the proof for the cases (i) and (ii).

In case (iii) with $\lambda=m+2s$, we consider instead all the horizontal derivatives of $u$ and use the inductive hypothesis (A.33) in the form

[TABLE]

where $q_{m-2k}^{i}$ are $(m-1-2k)$-homogeneous polynomials. Therefore,

[TABLE]

with $(m-2k)q_{m-2k}(x)=\sum_{i=1}^{n}q_{i,m-1-2k}(x^{\prime})x_{i}$. Taking into account the homogeneity of $u$, we infer that $u(0,x_{n+1})=c|x_{n+1}|^{m+2s}$ and the exact form for the polynomials $q_{k}$ in (A.33) follows by using the equation (A.25).

Finally, we discuss the case of solution to the obstacle problem (1.1). In case (i), the unilateral condition $u\geq 0$ on $B_{1}^{\prime}$ implies that

[TABLE]

This implies that the polynomials $p_{k}$ with $k\in\{1,\ldots,m\}$ are all zero. Let, indeed, $j:=\min\{k\in\{1,\ldots,m\}:\,p_{k}\not\equiv 0\}$ and divide $u$ by $x_{n}^{j}>0$: by taking the limit as $x_{n}\downarrow 0$ we infer that $p_{j}$ is a constant sign homogeneous harmonic polynomial, which holds only if $p_{j}\equiv 0$, thus giving a contradiction. Therefore, we conclude $u(x)=p_{0}\Phi_{m}(x_{n},x_{n+1})$ with $p_{0}>0$ for solutions to the obstacle problem.

For the case (ii), by the same argument we deduce that all polynomials $p_{k}^{\pm}\equiv 0$ for $k\in\{1,\ldots,m\}$, and therefore $u(x)=p^{+}_{0}\Psi_{m}(x_{n},x_{n+1})+p^{-}_{0}\Psi_{m}(-x_{n},x_{n+1})$ with $p_{0}^{\pm}\geq 0$. Since $u$ is a function of two variables, the conclusion follows now from Lemma 5.3. ∎