Unique continuation principles in cones under nonzero Neumann boundary conditions

TL;DR

This paper establishes unique continuation principles for elliptic equations in conical domains with Neumann boundary conditions, accommodating singularities and providing classification of blow-up limits.

Contribution

It introduces novel unique continuation results for elliptic equations in cones with nonzero Neumann conditions, even with singularities at the vertex.

Findings

Unique continuation results for elliptic equations in cones

Classification of blow-up limits in this setting

Use of Almgren-type frequency formula with remainders

Abstract

We consider an elliptic equation in a cone, endowed with (possibly inhomogeneous) Neumann conditions. The operator and the forcing terms can also allow non-Lipschitz singularities at the vertex of the cone. In this setting, we provide unique continuation results, both in terms of interior and boundary points. The proof relies on a suitable Almgren-type frequency formula with remainders. As a byproduct, we obtain classification results for blow-up limits.

Equations

$$\Sigma =\left\{\frac{x}{|x|}:x\in \Omega \right\}\subset {\mathbb{S}}^{n-1}$$

$$c\leqslant A(x)\leqslant\frac{1}{c}\quad{\mbox{ for some $c>0$ and a.e. $x\in\Omega$.}}$$

$$\begin{cases}{\rm div}\,\big{(}A(x)\,\nabla u(x)\big{)}=g(x,u(x)),&{\mbox{ for every }}x\in\Omega\cap B_{1},\\ A(x)\nabla u(x)\cdot\nu(x)=f(x,u(x)),&{\mbox{ for every }}x\in B_{1}\cap\partial\Omega,\end{cases}$$

$${\int }_{B_1\cap \Omega }A(x)\nabla u(x)\cdot \nabla \phi (x)dx=-{\int }_{B_1\cap \Omega }g(x,u(x))\phi (x)dx+{\int }_{B_1\cap \partial \Omega }f(x,u(x))\phi (x)d{\mathcal{H}}_x^{n-1}.$$

$$\begin{array}{rl}D(r)& :=r^{2-n}{\int }_{B_r\cap \Omega }A(x)|\nabla u(x){|}^{2}dx-r^{2-n}{\int }_{B_r\cap \partial \Omega }f(x,u(x))u(x)d{\mathcal{H}}_x^{n-1}\\ & +r^{2-n}{\int }_{B_r\cap \Omega }g(x,u(x))u(x)dx\\ \text{mbox}andH(r):& =r^{1-n}{\int }_{\partial B_r\cap \Omega }A(x)u^2(x)d{\mathcal{H}}_x^{n-1}\\ & ={\int }_{\Sigma }A(ry)u^2(ry)d{\mathcal{H}}_y^{n-1}.\end{array}$$

$$\mathcal{N}(r):=\frac{D(r)}{H(r)}.$$

$$|\nabla A(x)\cdot x|⩽{\epsilon }_{r}A(x),$$

$$|\nabla A(x)|⩽\frac{CA(x)}{|x|},$$

$$|f(x,t)|⩽CA(x)|x{|}^{\delta -1}|t|,$$

$$|{\nabla }_{x}f(x,t)|⩽CA(x)|x{|}^{\delta -2}|t|,$$

$$|g(x,t)|⩽CA(x)|x{|}^{\delta -2}|t|,$$

$$F(x,t):={\int }_0^{t}f(x,\tau )d\tau .$$

$$u\in H^1(\Omega \cap B_1)\cap L^{\infty }(\Omega \cap B_1)$$

$$u\not\equiv 0$$

$$H(r)>0\text{and}\mathcal{N}(r)+1>0\text{for all r∈(0,r0)};$$

$${\mathcal{N}}^{\prime }(r)⩾-C_1\max \{r^{\delta },{\epsilon }_r\}{r}^{-1}(2+\mathcal{N}(r))\text{for all }r\in (0,r_1).$$

$$r\mapsto \frac{{\epsilon }_r}{r}\in L^1(0,r_1),$$

$$\gamma :=\underset{r\searrow 0}{\lim }\mathcal{N}(r)$$

$$\underset{\Omega \ni x\to 0}{\lim }\frac{u(x)}{|x{|}^k}=0.$$

$$u\equiv 0$$

$$u_{\lambda }(x):=\frac{u(\lambda x)}{\sqrt{H(\lambda )}}.$$

$$|f_{t}(x,t)|\leqslant C\,|x|^{\delta-1},\quad\mbox{for a.e. $x\in\Omega\cap B_{1}$ and any $t\in\mathbb{R}$},$$

$$\underset{x\to 0}{\lim }A(x)=1.$$

$$\tilde{u}(x)=|x{|}^{\gamma }\psi \left(\frac{x}{|x|}\right),$$

$$\gamma =-\frac{n-2}{2}+\sqrt{{\left(\frac{n-2}{2}\right)}^2+{\lambda }_{k_0}(\Sigma )}⩾0$$

$${\int }_{\Sigma }{\psi }^2(x)d{\mathcal{H}}_x^{n-1}=1.$$

$$\underset{\partial \Omega \ni x\to 0}{\lim }\frac{u(x)}{|x{|}^k}=0.$$

$$u\equiv 0$$

$$\nu (x)\cdot x=0\text{mbox}foranyx\in \partial \Omega \setminus \{0\}.$$

$$0={\Phi }_0(x_0)={\Phi }_0(t{x}_0),$$

Full text

In this setting, we provide unique continuation results, both in terms of interior and boundary points.

The proof relies on a suitable Almgren-type frequency formula with remainders. As a byproduct, we obtain classification results for blow-up limits.

The authors are member of INdAM/GNAMPA. S. Dipierro is supported by the Australian Research Council DECRA DE180100957 “PDEs, free boundaries and applications” and the Fulbright Foundation. S. Dipierro and E. Valdinoci are supported by the Australian Research Council Discovery Project DP170104880 NEW “Nonlocal Equations at Work”. V. Felli is partially supported by the PRIN-2015 grant “Variational methods, with applications to problems in mathematical physics and geometry”. This work was started on the occasion of a very fruitful visit of V. Felli to the University of Melbourne.

1. Introduction

In this article we consider an elliptic equation with Neumann boundary condition. The domain taken into consideration is a cone, and the equation and the boundary condition can be inhomogeneous and be singular at the origin.

The main results that we provide are of unique continuation type. Roughly speaking, we will show that if a solution vanishes at any order at the vertex of the cone, then the solution must necessarily vanish in a neighborhood of the vertex (and then everywhere, up to suitable assumptions).

The notion of vanishing can be framed both with respect to the convergence of points coming from the interior of the domain and, under the appropriate assumptions, with respect to the convergence of points coming from the boundary.

From these results, we also obtain classification results for the blow-up limits. The method of proof will rely on the special geometric structure of the cone, which is a set invariant under dilations and in which the normal on the side of the cone is perpendicular to the radial direction. The main analytic tool in use will be an appropriate type of frequency function. Differently from the classical case in [MR574247], the choice of the frequency function in our case has to comprise additional quantities and reminders to deal with the forcing terms and possibly compensate for the singular behaviors near the vertex.

The mathematical setting in which we work is the following. We let $\Omega\subseteq\mathbb{R}^{n}$, with $n\geqslant 2$, be a cone with vertex at the origin (namely, we assume that $x\in\Omega$ if and only if $tx\in\Omega$ for all $t>0$). We consider the spherical cap

[TABLE]

and we assume that $\Sigma$ has $C^{2}$ boundary in ${\mathbb{S}}^{n-1}$.

We also take into account a positive function $A\in W^{1,1}(\Omega)$ such that

[TABLE]

For every $r>0$ we denote $B_{r}=\{x\in\mathbb{R}^{n}:|x|<r\}$. We deal with weak solutions of the following partial differential equation in a neighbourhood of the vertex of the cone (to fix the notations we consider $\Omega\cap B_{1}$) with possibly inhomogeneous Neumann datum:

[TABLE]

where $\nu(x)$ denotes the exterior unit normal of $\Omega$ at $x\in\partial\Omega$, $f\in C^{1}((\overline{\Omega}\setminus\{0\})\times\mathbb{R})$, and $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function.

We say that a function $u\in H^{1}(B_{1}\cap\Omega)$ is a weak solution to (1.3) if, for all $\varphi\in C^{\infty}_{\rm c}(B_{1}\cap\overline{\Omega})$,

[TABLE]

As a technical observation, we point out that the integrals at the right hand side of the above identity are finite under the assumptions of Theorem 1.1 below in view of the Poincaré-type Inequality and the Trace Inequality proved in Corollary 2.3 and Lemma 2.5 respectively.

The use of Almgren-type frequency functions to study unique continuation properties of elliptic partial differential equations dates back to the pioneering contribution of Garofalo and Lin [MR833393] and relies essentially on the possibility of deducing from the boundedness of the frequency quotient a doubling-type condition. Unique continuation from boundary points was investigated via Almgren-type monotonicity arguments in [MR1466583, MR1363203, MR3109767, MR1415331, MR2370633]. As far as elliptic equations with Neumann-type boundary conditions are concerned, we mention that in [MR2162295] boundary unique continuation theorems and doubling properties near the boundary were established under zero Neumann boundary conditions. The main novelty of the present paper is a strong unique continuation result for solutions whose restriction to the boundary vanishes at any order at the vertex under non-homogeneous Neumann boundary conditions, while in [MR2162295, Theorem 1.7] unique continuation from the boundary was proved for solutions vanishing on positive surface measure subsets of the boundary and satisfying a zero Neumann condition on such set. The achievement of such a result requires a combination of the monotonicity argument with a blow-up analysis for scaled solutions, in the spirit of [MR2735078, MR3169789].

We now introduce the notation needed to define the frequency function for our setting. For $r>0$, we define

[TABLE]

We also introduce the “Almgren frequency function” in our framework, given by

[TABLE]

With this setting, the pivotal result that we obtain is an appropriate monotonicity formula with reminders, which we state as follows:

Theorem 1.1**.**

Suppose that (1.2) holds and

[TABLE]

for some $C>0$ and $\delta>0$.

Let also

[TABLE]

Let

[TABLE]

be a solution of (1.3) in the sense of (1.4), such that

[TABLE]

for all $r\in(0,1)$.

Then the following holds true.

1. (i)

There exists $r_{0}>0$ such that

[TABLE]

in particular the function $\mathcal{N}$ defined in (1.6) is well defined on $(0,r_{0})$. 2. (ii)

There exist $r_{1}\in(0,r_{0})$ and $C_{1}>0$ such that

[TABLE] 3. (iii)

If also

[TABLE]

then the limit

[TABLE]

exists, is finite and $\gamma\geqslant 0$.

We observe that the assumptions of Theorem 1.1 are very general and do not necessarily require the weight $A$ to be Lipschitz continuous or the source terms $f$ and $g$ to be bounded. In particular, estimate (1.16) requires assumptions (1.7) and (1.8) which could be satisfied even by unbounded potentials, as for example $A(x)=\log|x|(\cos(x_{n}/|x|)-2)$. On the other hand, to prove that $\mathcal{N}$ is bounded and has finite limit as $r\to 0^{+}$ assumption (1.17) is also needed; we observe that (1.17) forces the boundedness of $A$ but could be satisfied by non-Lipschitz continuous weights, like $A(x)=1+|x|^{\delta}$ with $\delta$ positive and small, for example.

The functions $f$ and $g$ can be singular as well, in accordance with (1.9) and (1.11). To allow all these possible singularities, it is crucial that the “frequency function” also takes into account the special behaviors of $A$, $f$ and $g$, as in (1.5). Moreover, the special geometry of the cone $\Omega$ will turn out to be the cornerstone for our main estimates to hold, thus providing an interesting interplay between analytic and geometric properties of the problem.

We also observe that condition (1.14) is quite natural, since it requires that the solution is nontrivial in any neighborhood of the vertex of the cone. Furthermore, under the additional assumption that $A$ is locally Lipschitz continuous, assumption (1.14) is satisfied by all nontrivial solutions, in light of the classical unique continuation principle in [MR882069], see also [MR1233189] (similarly, if $A$ satisfies a Muckenhoupt-type assumption, then (1.14) is a consequence of the unique continuation principle in [MR2370633], see also [MR833393]).

From Theorem 1.1 and a “doubling property” method one obtains a number of results of unique continuation type. In this spirit, we first provide a unique continuation result from the vertex of the cone with respect to interior points:

Theorem 1.2**.**

Let $u$ be a solution of (1.3), under assumptions (1.2), (1.7), (1.8), (1.9), (1.10), (1.11), (1.13) and (1.17).

Assume also that $u$ vanishes at the origin at any order with respect to interior points, namely that for every $k\in\mathbb{N}$

[TABLE]

Then there exists $r>0$ such that

[TABLE]

If, in addition, $A$ is locally Lipschitz continuous, then

[TABLE]

An interesting consequence of our Theorem 1.1 deals with blow-up limits. Namely, for each $\lambda>0$, we define

[TABLE]

We consider the Laplace-Beltrami operator ${\mathcal{L}}_{\Sigma}:=-\Delta_{\mathbb{S}^{n-1}}$ on the spherical cap $\Sigma$ under null Neumann boundary conditions. By classical spectral theory, the spectrum of the operator ${\mathcal{L}}_{\Sigma}$ is discrete and consists in a nondecreasing diverging sequence of eigenvalues $0=\lambda_{1}(\Sigma)<\lambda_{2}(\Sigma)\leqslant\cdots\leqslant\lambda_{k}(\Sigma)\leqslant\cdots$ with finite multiplicity.

In the following theorem we describe the limit profiles of the blowed-up family (1.22) in terms of the eigenvalues and the eigenfunctions of $\mathcal{L}_{\Sigma}$.

Theorem 1.3**.**

Let $u$ be a solution of (1.3), under assumptions (1.2), (1.7), (1.8), (1.9), (1.10), (1.11), (1.13) and (1.17).

Assume that (1.14) holds true,

[TABLE]

and that

[TABLE]

Then, up to a subsequence, as $\lambda\searrow 0$, we have that $u_{\lambda}$ converges strongly in $H^{1}(\Omega\cap B_{1})$ to a function $\tilde{u}$ which is positively homogeneous and can be written in the form

[TABLE]

where

[TABLE]

for some $k_{0}\in\mathbb{N}\setminus\{0\}$ and $\psi$ is an eigenfunction of the operator $\mathcal{L}_{\Sigma}$ associated to the eigenvalue $\lambda_{k_{0}}(\Sigma)$ such that

[TABLE]

From Theorem 1.3, one can also obtain a unique continuation result from the vertex of the cone with respect to boundary points:

Theorem 1.4**.**

Let $u$ be a solution of (1.3), under assumptions (1.2), (1.7), (1.8), (1.9), (1.10), (1.11), (1.13), (1.17), (1.23) and (1.24).

Assume also that $u$ vanishes at the origin at any order with respect to boundary points, namely that for every $k\in\mathbb{N}$

[TABLE]

Then there exists $r>0$ such that

[TABLE]

If, in addition, $A$ is locally Lipschitz continuous, then

[TABLE]

We stress that while (1.19) is assumed for interior points, we have that hypothesis (1.27) focuses on boundary points.

The rest of the article is organized as follows. Section 2 presents a number of ancillary results, to be exploited in the proofs of the main theorems. In particular, we will collect there some observations on the geometry of the cone and suitable functional inequalities.

The proof of Theorem 1.1 is presented in Section 3 and will serve as a pivotal result for the main theorems of this paper. Namely, Theorem 1.2 will be proved in Section 4, Theorem 1.3 will be proved in Section 5, and Theorem 1.4 will be proved in Section 6.

2. Toolbox

This section collects ancillary results used in the main proofs.

2.1. Cone structure

We recall here an elementary property of the cones:

Lemma 2.1**.**

Let $\Omega\subset\mathbb{R}^{n}$ be a cone with respect to the origin. Then

[TABLE]

Proof.

Fixed $x_{0}\in\partial\Omega\setminus\{0\}$, we have that there exists $r_{0}>0$ such that $\Omega\cap B_{r}(x_{0})$ coincides with the sublevel sets of some nondegenerate function $\Phi_{0}:B_{r}(x_{0})\to\mathbb{R}$, with $\nu(x)=\frac{\nabla\Phi_{0}(x)}{|\nabla\Phi_{0}(x)|}$. By the cone structure of $\Omega$, we thereby see that, for any $t$ close to $1$,

[TABLE]

and so

[TABLE]

This proves that $\nu(x_{0})\cdot x_{0}=0$ and establishes (2.1). ∎

2.2. A Poincaré-type Inequality

In this subsection, we provide some results concerning suitable weighted Poincaré-type Inequalities which will play an important role in some of the technical estimates needed to prove the main results.

Lemma 2.2**.**

Let $\mu\in(-\infty,n)$. Let $\Omega\subset\mathbb{R}^{n}$ be a cone with respect to the origin such that the spherical cap $\Sigma$ defined in (1.1) is smooth. Let $A\in\L^{\infty}(\Omega)$ satisfy (1.8). For every $r>0$ and $u\in C^{\infty}(\overline{\Omega\cap B_{r}})$

[TABLE]

Proof.

Let $u\in C^{\infty}(\overline{\Omega\cap B_{r}})$. Since

[TABLE]

by the Divergence Theorem and (2.1) we deduce that

[TABLE]

and hence the conclusion follows. ∎

Corollary 2.3**.**

Let $\mu\in(-\infty,n)$. Let $\Omega\subset\mathbb{R}^{n}$ be a cone with respect to the origin such that the spherical cap $\Sigma$ defined in (1.1) is smooth. Let $c\in\left(0,\frac{n-\mu}{2}\right)$ and $A\in\L^{\infty}(\Omega)$ satisfy (1.8) and (1.7). Then there exists $r_{\mu}>0$ such that for every $r\in(0,r_{\mu})$ and $u\in C^{\infty}(\overline{\Omega\cap B_{r}})$

[TABLE]

Proof.

Exploiting (1.7), we observe that

[TABLE]

as long as $r$ is small enough, and hence the desired result follows by Lemma 2.2. ∎

For $\mu<2$ the previous corollary yields the following result.

Corollary 2.4**.**

Let $\mu<2$. Let $\Omega\subset\mathbb{R}^{n}$ be a cone with respect to the origin such that the spherical cap $\Sigma$ defined in (1.1) is smooth. Let $c\in\left(0,\frac{n-\mu}{2}\right)$ and $A\in\L^{\infty}(\Omega)$ satisfy (1.8) and (1.7). Then there exists $r_{\mu}>0$ such that, for every $r\in(0,r_{\mu})$ and $u\in H^{1}(\Omega\cap B_{r})$, $u|x|^{-\mu/2}\in L^{2}(\Omega\cap B_{r})$ and

[TABLE]

Proof.

The inequality for $u\in C^{\infty}(\overline{\Omega\cap B_{r}})$ follows esily from Corollary 2.3 and the fact that, since $2-\mu>0$, $|x|^{2-\mu}\leqslant r^{2-\mu}$ in $\Omega\cap B_{r}$. The conclusion follows by density and the Fatou’s Lemma. ∎

2.3. Trace Inequalities

Now we present a result of trace-type which will be exploited in the proofs of the main theorems.

Lemma 2.5**.**

Let $\gamma\in(-\infty,n-1)$. Let $\Omega\subset\mathbb{R}^{n}$ be a cone with respect to the origin such that the spherical cap $\Sigma$ defined in (1.1) is smooth. Let $A\in\L^{\infty}(\Omega)$ satisfy (1.2). For every $r>0$ and $u\in C^{\infty}(\overline{\Omega\cap B_{r}})$ we have that

[TABLE]

for some $C>0$ independent of $r$. Furthermore, if $\gamma<1$, then every function $u\in H^{1}(\Omega\cap B_{r})$ has a trace belonging to $L^{2}(\partial\Omega\cap B_{r};|x|^{-\gamma/2})$ and

[TABLE]

Proof.

We let $u\in C^{\infty}(\overline{\Omega\cap B_{r}})$. Also, for all $\rho\in(0,r)$ and $\theta\in\Sigma$, we define $u^{(\rho)}(\theta):=u(\rho\theta)$. By Fubini’s Theorem and the Sobolev Trace Theorem on manifolds we have that

[TABLE]

where $\nabla_{\theta}$ denotes the tangential gradient along $\Sigma$, so that, if $x=\rho\theta$,

[TABLE]

Hence, in view of (1.2), we find that

[TABLE]

which yields the inequality for functions in $C^{\infty}(\overline{\Omega\cap B_{r}})$. If $\gamma<1$, then $|x|^{1-\gamma}\leqslant r^{1-\gamma}$ in $\Omega\cap B_{r}$, then The conclusion follows by density and the Fatou’s Lemma. ∎

3. Proof of Theorem 1.1

We first observe that, by elliptic regularity theory (see e.g. Theorem 8.13 in [MR2399851], [MR0125307, MR0162050] or [MR0350177]) we have that, under the assumptions of Theorem 1.1,

[TABLE]

We denote by $\nu$ both the exterior normal at $\partial\Omega$ and the exterior normal at $\partial B_{r}$, since no confusion can arise. Testing the equation in (1.3) against the solution itself, we see that

[TABLE]

Hence, recalling (1.5),

[TABLE]

Using again (1.3), we also observe that

[TABLE]

On the other hand, from (1.5) we know that

[TABLE]

and (recalling that $\Omega$ is a cone, hence $\Omega/r=\Omega$ for each $r>0$)

[TABLE]

Thus, comparing (3.2) with (3.5) we conclude that

[TABLE]

and therefore

[TABLE]

From (3.1) it follows that, for all $0<\delta<r<1$, $A(\nabla u\cdot x)\nabla u-\frac{A}{2}|\nabla u|^{2}x\in W^{1,1}(\Omega\cap(B_{r}\setminus B_{\delta})$ so that

[TABLE]

Since

[TABLE]

there exists a decreasing sequence $\{\delta_{n}\}\subset(0,1)$ such that $\lim_{n\to+\infty}\delta_{n}=0$ and

[TABLE]

Choosing $\delta=\delta_{n}$ in (3.7) and letting $n\to\infty$ we then obtain

[TABLE]

Therefore, taking into account (3.3),

[TABLE]

We thereby substitute this identity into (3.4) and we conclude that

[TABLE]

From this and (3.6), we find that

[TABLE]

On the other hand, recalling (3.5), we see that

[TABLE]

Hence, substituting this identity into (3.8), we conclude that

[TABLE]

Moreover, from the Cauchy-Schwarz Inequality, we know that

[TABLE]

Consequently, using again (1.5), we also observe that

[TABLE]

Plugging this information into (3.9), we thus obtain that

[TABLE]

Then, from (3.10) and (2.1), we obtain that

[TABLE]

Now, we define

[TABLE]

By (1.9), we have that

[TABLE]

On the other hand, by Lemma 2.5 (used here with $\gamma:=1-\delta$), we see that

[TABLE]

Hence, in view of Corollary 2.4 (used here with $\mu:=2-\delta$), (1.5) and (3.12)

[TABLE]

Therefore, in light of (3.13)

[TABLE]

Also, by (1.11) and Corollary 2.4 (used here with $\mu:=2-\delta$),

[TABLE]

Consequently, by (3.15) and (3.16)

[TABLE]

and therefore, for any $r\in(0,1)$ sufficiently small,

[TABLE]

Estimate (3.17) implies statement (i) with $r_{0}>0$ so small as to satisfy condition (3.17) and $Cr_{0}^{\delta}<1$. Indeed, let us argue by contradiction and assume that there exists $\bar{r}\in(0,r_{0})$ such that $H(\bar{r})=0$. By (1.5) this would imply that $u\equiv 0$ on $\Omega\cap\partial B_{\bar{r}}$ and hence, in view of (3.2), $D(\bar{r})=0$. Then (3.17) yields that $E(\bar{r})=0$ and hence $u$ is constant in $\Omega\cap B_{\bar{r}}$. Therefore $u\equiv 0$ in $\Omega\cap B_{\bar{r}}$, which is in contradiction with (1.14).

Furthermore, for all $r\in(0,r_{0})$, (3.17) implies that

[TABLE]

and hence $\mathcal{N}(r)+1>0$.

Moreover, from the Sobolev Trace Theorem on manifolds applied on the spherical cap $\partial B_{r}\cap\partial\Omega=r\partial\Sigma$, we have that, recalling (1.2),

[TABLE]

for some $C>0$ independent of $r$ (varying from line to line). Now, we recall (1.9) and we observe that

[TABLE]

In addition, from (1.11),

[TABLE]

From (3.4), (3.20) and (3.21), we obtain that

[TABLE]

Then from (3.19) it follows that

[TABLE]

from which it follows that

[TABLE]

for some $C>0$ and for all $r>0$ sufficiently small.

Plugging (3.22) into (3.19) we conclude that

[TABLE]

as long as $r$ is sufficiently small. It is now our goal to use the previously obtained information in order to estimate the right hand side of (3.11). To this end, we first observe that, from (1.7),

[TABLE]

This and (3.22) lead to

[TABLE]

Furthermore, by (1.7) and (1.11),

[TABLE]

Consequently, exploiting Corollary 2.4 with $\mu:=2(1-\delta)$,

[TABLE]

Now, plugging the latter inequality, (3.15), (3.16), (3.21) and (3.24) into (3.11), we conclude that

[TABLE]

for some $C>0$.

Now, recalling (3.20) and (3.23), we notice that

[TABLE]

This and (3.25) give that

[TABLE]

Now, we denote by $\partial_{\star}:=\frac{x}{|x|}\cdot\nabla$ and we observe that $\partial_{\star}$ is the “radial” component of the tangential gradient along $\partial\Omega$, since $\Omega$ is a cone. Hence, since, by (1.12),

[TABLE]

we obtain that

[TABLE]

As a consequence, by (1.10),

[TABLE]

Moreover, integrating by parts along $\partial\Omega$,

[TABLE]

In addition, by (1.9) and (1.12), we know that

[TABLE]

This and (3.28) lead to

[TABLE]

Hence, recalling (3.27),

[TABLE]

up to renaming $C>0$.

Therefore, recalling (3.14) and (3.23),

[TABLE]

Then, we insert this information into (3.26) and we conclude that

[TABLE]

Accordingly, by (1.6),

[TABLE]

From this inequality and (3.17) we find that

[TABLE]

Let

[TABLE]

In view of (3.6), (1.15), and (1.7), for $r\in\Lambda$ we can estimate $D^{\prime}(r)$ as follows:

[TABLE]

It follows that, for all $r\in\Lambda$,

[TABLE]

Combining the previous estimate with (3.29) we obtain that, for all $r\in\Lambda$ sufficiently small

[TABLE]

For $r\not\in\Lambda$ estimate (3.30) is trivial, since the left hand side of (3.30) is nonnegative outside $\Lambda$ whereas the right hand side is nonpositive because of (1.15). Estimate (1.16) and statement (ii) are thereby proved.

To prove statement (iii), let $h(r):=\max\{r^{\delta},\varepsilon_{r}\}r^{-1}$. By assumption (1.17), we have that $h\in L^{1}(0,r_{1})$. Then, from (1.16) it follows that

[TABLE]

hence the function $w(r):=(2+\mathcal{N}(r))e^{-C_{1}\int_{r}^{1}h(s)\,ds}$ is nondecreasing in $(0,r_{1})$.

Moreover $w\geqslant 0$ in view of (1.15). Therefore $w$ admits a finite limit as $r\to 0^{+}$ and then also $\mathcal{N}$ has a finite limit $\gamma$ as $r\to 0^{+}$. Since estimate (3.17) implies that $\mathcal{N}(r)\geqslant-Cr^{\delta}$ in $(0,r_{0})$, we conclude that $\gamma\geqslant 0$.

4. Proof of Theorem 1.2

We start by proving (1.20). To this end, we argue for a contradiction and we suppose that (1.20) is violated. Then, we have that (1.14) is satisfied and hence all the hypotheses of Theorem 1.1 are fulfilled. In particular, by the fact that the limit in (1.18) is finite and $\mathcal{N}$ is continuous in $(0,r_{0})$, we find that $\mathcal{N}$ is bounded, i.e. for all $r\in(0,r_{0})$,

[TABLE]

for some $C>0$.

Moreover, by (3.6),

[TABLE]

As a consequence, recalling (1.8),

[TABLE]

for some $C>0$ independent of $r$ (varying from line to line). This and (4.1) yield that

[TABLE]

up to renaming $C>0$ and therefore, if $r\in(0,r_{0}/2)$,

[TABLE]

up to renaming $C$ line after line. More in general, integration of (4.2) over the interval $(r,rR)$ yields that for every $R>1$ there exists $C_{R}>0$ (depending on $R$ but independent of $r$) such that

[TABLE]

The inequality in (4.3) provides a pivotal “doubling property” in our setting. From this, we obtain that

[TABLE]

up to renaming $C$.

Integrating the latter inequality in $r$, we find that

[TABLE]

for some $C_{0}>0$ independent of $r$, which gives that

[TABLE]

for all $m\in\mathbb{N}$ and $r\in(0,r_{0})$.

Now we fix $k\in\mathbb{N}$ such that $2^{2k}\geqslant 2C_{0}$. In light of (1.19) we can write that

[TABLE]

as long as $x\in\Omega$ and $|x|$ is sufficiently small. Hence, we can exploit (4.5) for $m$ sufficiently large and conclude that

[TABLE]

Then, sending $m\to+\infty$, we conclude that

[TABLE]

and therefore, by (1.2), it follows that $u$ must vanish necessarily in $B_{r_{0}}\cap\Omega$. This proves (1.20), against our initial contradictory assumption.

Having established (1.20), we can now complete the proof of Theorem 1.2, since, if $A$ is Lipschitz, we can use (1.20) and the classical unique continuation principle in [MR882069] and obtain (1.21), as desired.

5. Proof of Theorem 1.3

By (1.3) and (1.22), we see that, if $x\in\Omega$ and $\lambda$ is sufficiently small,

[TABLE]

where

[TABLE]

Similarly, we see that, if $x\in\partial\Omega$,

[TABLE]

where

[TABLE]

Now, in the notation of (1.5), we write $D_{u,A,f,g}$ and $H_{u,A}$ to emphasize their dependences. In the same way, in the notation of (1.6), we write ${\mathcal{N}}_{u,A,f,g}$. For short, we drop the indexes when they refer to the original configuration in (1.3) and we write

[TABLE]

We remark that

[TABLE]

In addition,

[TABLE]

and therefore

[TABLE]

This and (1.18) give that, for all $r>0$,

[TABLE]

for some finite $\gamma\geqslant 0$.

Now we claim that, for all $R>0$ and $\lambda\in(0,r_{0}/R)$,

[TABLE]

for some $C_{R}>0$ (eventually depending on $R$). To this end, we exploit (3.12), (3.18), (4.4), and (4.1) to see that, for all $\lambda\in(0,r_{0}/R)$,

[TABLE]

for some $C_{R}>0$ depending on $R$.

Moreover, using again (4.4), we observe that

[TABLE]

up to renaming $C_{R}$. Hence, recalling Corollary 2.4 (used here with $\mu:=0$, $r:=R$, and on the function $u_{\lambda}$ and with weight $A_{\lambda}$) and (5.6),

[TABLE]

up to renaming $C_{R}$. This inequality and (5.6), combined with (1.2), give (5.5), as desired.

Now, from (5.5) and a diagonal process, we deduce that, along a subsequence, $u_{\lambda}$ converges a.e. in $\Omega$, strongly in $L^{2}(\Omega\cap B_{R})$ and weakly in $H^{1}(\Omega\cap B_{R})$ for all $R>0$, as $\lambda\searrow 0$. Consistently with the notation in Theorem 1.3, we denote by $\tilde{u}$ this limit; we observe that $\tilde{u}\in\bigcap_{R>0}H^{1}(\Omega\cap B_{R})$.

As a particular case of (5.7) with $R=1$ we have that

[TABLE]

which, in view of the compactness of the trace embedding $H^{1}(\Omega\cap B_{1})\hookrightarrow\hookrightarrow L^{2}(\Omega\cap\partial B_{1})$, implies that

[TABLE]

Hence $\tilde{u}\not\equiv 0$.

We observe that, by (1.11), for every $x\in B_{1}$,

[TABLE]

up to renaming $C$ line after line.

Moreover, by (1.9),

[TABLE]

Now we claim that, for all $R>0$,

[TABLE]

Indeed, using (5.10), Corollary 2.4 (used here with $A:=1$, $r:=R$, and $\mu:=2-\delta$), and (1.2), we see that

[TABLE]

From this, (5.7) and (5.6), we deduce that

[TABLE]

up to renaming $C_{R}>0$.

This proves the first claim in (5.12), and we now prove the second. For this, using (5.11), and then Lemma 2.5 (with $A:=1$, $r:=R$ and $\gamma:=1-\delta$) we find that

[TABLE]

Hence, using Corollary 2.4 as before, we obtain that

[TABLE]

which implies the second claim in (5.12). This completes the proof of (5.12).

Now we claim that

[TABLE]

To this end, we exploit (5.1) and (5.2) and, given $\varphi\in C^{\infty}_{0}(\mathbb{R}^{n})$, we write that

[TABLE]

Hence, in light of (1.24), (5.10) and (5.11),

[TABLE]

where $C^{\prime}$, $R>0$ may also depend on $\varphi$. Consequently, using Corollary 2.4 and Lemma 2.5 as before, we obtain

[TABLE]

up to renaming $C^{\prime}>0$, that is

[TABLE]

Since this identity holds true for all $\varphi\in C^{\infty}_{0}(\mathbb{R}^{n})$, we have completed the proof of (5.13).

We now show that

[TABLE]

for some $C>0$. To accomplish this, we will exploit elliptic regularity theory, see e.g. Theorem 8.13 in [MR2399851] (with the notation in Example 6.2 on page 314 in [MR2399851] for the definition of the norms) or [MR0125307, MR0162050] and Theorem 5.1 in [MR0350177], considering a set $\Omega_{1}$ with smooth boundary and such that $\Sigma\subset\Omega_{1}\subset\Omega\cap(B_{2}\setminus B_{1/2})$. In this way, by (5.1) and (5.2),

[TABLE]

Moreover, in light of (5.8) and (5.10),

[TABLE]

Similarly, recalling (5.11) and (5.8),

[TABLE]

Furthermore, from (1.10), (1.23), and (5.5) it follows that

[TABLE]

Therefore

[TABLE]

which, in view of the continuous trace embedding $H^{1}(\Omega\cap(B_{2}\setminus B_{1/2}))\hookrightarrow H^{1/2}(\partial\Omega\cap(B_{2}\setminus B_{1/2}))$, yields

[TABLE]

up to renaming $C$. From this, (5.8), (5.16) and (5.15), we conclude that

[TABLE]

again up to renaming $C>0$. Thus, using the trace embedding,

[TABLE]

up to renaming $C>0$, and consequently, up to a subsequence, we obtain that

[TABLE]

Now we notice that, exploiting (5.1) and (5.2),

[TABLE]

Using this, (1.24), (5.12) and (5.17), we conclude that

[TABLE]

Hence, recalling (5.13),

[TABLE]

Since the weak convergence and the convergence of the norm imply the strong convergence in $L^{2}(\Omega\cap B_{1})$, we thereby conclude that $\nabla u_{\lambda}$ converges to $\nabla\tilde{u}$ strongly in $L^{2}(\Omega\cap B_{1},\mathbb{R}^{n})$, and this gives (5.14), as desired.

From (5.14) and (5.12), recalling (5.13) and the notation in (5.3), we conclude that

[TABLE]

As a consequence, exploiting (5.4),

[TABLE]

From this, we conclude that

[TABLE]

and hence we can write $\tilde{u}$ as in (1.25).

For completeness, we give a self-contained proof of (5.19) by arguing as follows. By (5.18), we know that ${\mathcal{N}}_{\tilde{u},1,0,0}^{\prime}(r)=0$, and therefore, by (1.6),

[TABLE]

Hence, exploiting (3.9) in this setting, and recalling (2.1), we see that

[TABLE]

By the Cauchy-Schwarz Inequality, the latter term is nonnegative, and consequently we find that $\tilde{u}$ is proportional to $\nabla\tilde{u}\cdot\nu$. Accordingly, we have that $\tilde{u}$ is a positively homogeneous function, of some degree $\gamma^{\prime}$.

Then, using (5.18) once again

[TABLE]

On the other hand, by (5.13),

[TABLE]

Plugging this information into (5.20), we thereby conclude that

[TABLE]

and then $\gamma^{\prime}=\gamma$. This completes the proof of (5.19) (and thus of (1.25)).

We also remark that, by (1.25) and (5.13), using the notation $\rho:=|x|$ and $\vartheta:=x/|x|$,

[TABLE]

and therefore $\psi$ is an eigenfunction of te operator $\mathcal{L}_{\Sigma}$; the Neumann boundary condition of $\psi$ also follows from the one of $\tilde{u}$ in (5.13).

Furthermore, by (5.9) and (1.25)

[TABLE]

which gives (1.26). The proof of Theorem 1.3 is thereby complete.

6. Proof of Theorem 1.4

First, we prove (1.28). We argue by contradiction, supposing that (1.28) does not hold, and therefore (1.14) is satisfied. Hence, we are in the position of using Theorem 1.3, and we let $\tilde{u}$ and $\psi$ as in (1.25). We note that, by (5.13) and elliptic regularity theory, we have that $\tilde{u}$ is smooth on $\overline{\Omega}\setminus\{0\}$.

We observe that the trace of $\tilde{u}$ on $B_{1}\cap\partial\Omega$ (which belongs to $L^{2}(B_{1}\cap\partial\Omega)$ by trace embeddings) cannot vanish identically, i.e.

[TABLE]

otherwise $\tilde{u}$ would be a harmonic function with homogeneous Dirichlet and Neumann conditions on $B_{1}\cap\partial\Omega$, and then necessarily $\tilde{u}$ would vanish identically in $B_{1}\cap\Omega$ (otherwise its trivial extension would violate classical unique continuation principles), in contradiction with (1.26).

From assumption (1.27) it follows that, for all $k\in\mathbb{N}$

[TABLE]

Since, in view of (1.22),

[TABLE]

and, by Theorem 1.3, $u_{\lambda}\to\tilde{u}$ in $L^{2}(B_{1}\cap\partial\Omega)$ along a subsequence, from (6.1) and (6.2) we conclude that

[TABLE]

for all $k\in\mathbb{N}$. Consequently, for all $k\in\mathbb{N}$, there exists $\lambda_{0}(k)\in(0,r_{0}/2)$ such that, for all $\lambda\in(0,\lambda_{0}(k)]$,

[TABLE]

On the other hand, by (4.3),

[TABLE]

for all $m\in\mathbb{N}$ and $\lambda\in(0,2^{-m}r_{0})$, for a suitable $C>0$ independent of $\lambda$ and $m$. This and (6.3) give that, for all $k$, $m\in\mathbb{N}$ and for all $\lambda\in(0,\min\{\lambda_{0}(k),2^{-m}r_{0}\})$,

[TABLE]

As a consequence, recalling (1.5) and integrating,

[TABLE]

for all $k$, $m\in\mathbb{N}$ and for all $\lambda\in(0,\min\{\lambda_{0}(k),2^{-m}r_{0}\})$.

We choose $m_{\lambda}\in\mathbb{N}$ such that

[TABLE]

so that $\lambda<2^{-m_{\lambda}}r_{0}$ for all $\lambda<\frac{r_{0}}{2}$. Then we find that

[TABLE]

for all $k\in\mathbb{N}$ and for all $\lambda\in(0,\lambda_{0}(k)]$.

Hence, since, by (6.4), we know that $2^{m_{\lambda}}\lambda\in\left[\frac{r_{0}}{2},r_{0}\right]$,

[TABLE]

for some suitable $\theta,\kappa>0$ depending only on $n,C,r_{0}$ (but independent of $k$), for all $k\in\mathbb{N}$ and for all $\lambda\in(0,\min\{\lambda_{0}(k),r_{0}/4\})$.

Accordingly, choosing $k\in\mathbb{N}$ sufficiently large such that $n+2k-\theta>0$ and sending $\lambda\searrow 0$, we conclude that

[TABLE]

This gives that (1.28) holds true, in contradiction with our initial hypothesis.

This completes the proof of (1.28). Finally, the proof of (1.29) is identical to the proof of (1.21), hence the proof of Theorem 1.4 is complete.