A geometric approach to regularity for nonlinear free boundary problems with finite Morse index

TL;DR

This paper establishes the continuous differentiability of free boundaries in certain nonlinear free boundary problems in two dimensions, especially for solutions with finite Morse index, using a novel geometric approach applicable to a broad class of nonlinearities.

Contribution

It introduces a geometric method to prove free boundary regularity for nonlinear problems with finite Morse index, extending beyond classical linear cases.

Findings

Free boundary is continuously differentiable in R^2 under finite connectivity.

Solutions with finite Morse index have free boundaries of finite connectivity.

Method applies to stationary points of a broad class of nonlinear functionals.

Abstract

Let $u$ be a weak solution of the free boundary problem $$\mathcal L u=\lambda_0 \mathcal H^1\lfloor\partial\{u>0\}, u\ge 0,$$ where $\mathcal L u={\text{div}}(g(\nabla u)\nabla u)$ is a quasilinear elliptic operator and $g(\xi)$ is a given function of $\xi$ satisfying some structural conditions. We prove that the free boundary $\partial\{ u>0\}$ is continuously differentiable in $\mathbb R^2$, provided that $\partial\{ u>0\}$ has locally finite connectivity. Moreover, we show that the free boundaries of weak solutions with finite $\it{Morse \ index}$ must have finite connectivity. The weak solutions are locally Lipschitz continuous and non-degenerate stationary points of the Alt-Caffarelli type functional $J[u]=\int_{\Omega}F(\nabla u)+Q^2\chi_{\{ u>0\}}$. The full regularity of the free boundary is not fully understood even for the {\it minimizers} of $J[u]$ in the simplest case $g(\xi)=|\xi|^{p-2}, p>1$, partly because the methods from the classical case $p=2$ cannot be generalized to the full range of $p$. Our method, however, is very geometric and works even for the $ stationary\ points$ of the functional $J[u]$ for a large class of nonlinearities $F$.

Equations

$$\mathcal{L}u={\lambda }_0{\mathcal{H}}^1\text{with}{\partial }_{\mathrm{red}}\{u>0\},u\ge 0,$$

$$\left\{\begin{array}[]{lll}\mathscr{L}u=0&\hbox{in}\ B_{r}(x_{0}),\\ |\nabla u|=\ell&\hbox{on}\ \partial\{{u}>0\}\cap B_{r}(x_{0}),\\ u\geq 0&\hbox{in}\ B_{r}(x_{0}),\\ \end{array}\right.$$

$$J[u]={\int }_{\Omega }F(|\nabla u|)+Q^2{\chi }_{\{u>0\}},u\in \mathcal{A}=\{v\in W^{1,F}(\Omega ),v-u_0\in W^{1,F}(\Omega )\},$$

$$F^{\prime }(\ell )\ell -F(\ell )=Q^2,$$

$$\mathcal{L}u={\lambda }_0{\mathcal{H}}^1\text{with}{\partial }_{\mathrm{red}}\{u>0\},$$

$$c_{\min }\le \frac{1}{r^n}{\int }_{\partial B_r(x)}u\le C_{\max },$$

$$-{\int }_{\Omega }\rho (\nabla u)\nabla u\nabla \zeta ={\lambda }_0{\int }_{{\partial }_{\mathrm{red}}\{u>0\}}\zeta d{\mathcal{H}}^{n-1},$$

$$\underset{\begin{array}{c}x\to x_0\\ x\in \{u>0\}\end{array}}{\mathrm{limsup}}|\nabla u(x)|=\ell .$$

$$\frac{|\{u=0\}\cap B_r(x)|}{|B_r(x)|}\ge \theta >0,\forall B_r(x)\subset B_1,x\in \partial \{u>0\}\cap B_1.$$

$$u_k\to u_0\text{ strongly in}{W}_{\mathrm{loc}}^{1,p}({\mathbb{R}}^2)\text{ and }{C}_{\mathrm{loc}}^{\alpha }({\mathbb{R}}^n),\forall \alpha \in (0,1)\text{as}k\to \infty ,$$

$$\partial \{u_k>0\}\to \partial \{u_0>0\}\text{in Hausdorff distance}{d}_{\mathcal{H}}\text{ locally in}{\mathbb{R}}^2,$$

$${\chi }_{\{u_k>0\}}\to {\chi }_{\{u_0>0\}}\text{in}{L}_{\mathrm{loc}}^1({\mathbb{R}}^2).$$

$$|\nabla u_0(x)|\le \ell ,x\in {\mathbb{R}}^2.$$

$$0$$

$${\mathcal{H}}^1(S)\le {\mathcal{H}}^1(S^{\prime }).$$

$$0$$

$$\widetilde{u}_{00}(x_{1},x_{2})=\left\{\begin{array}[]{lll}u_{00}(x_{1},x_{2})&\ \hbox{if}\ x_{1}\geq 0,\\ \ell x_{1}&\ \hbox{if}\ x_{1}<0,\end{array}\right.$$

$$\partial \{u_0>0\}\cap B_{\rho }\subset \left\{-\frac{{\bar{\sigma }}_0\rho }{2}<x\cdot e<\frac{{\bar{\sigma }}_0\rho }{2}\right\}\cap B_{\rho }$$

$$\partial \{u_k>0\}\cap B_{\rho }\subset \left\{-\frac{3{\bar{\sigma }}_0\rho }{4}<x\cdot e<\frac{3{\bar{\sigma }}_0\rho }{4}\right\}\cap B_{\rho }$$

$$x_0+\frac{1}{2}(E-x_0)\subset K\subset E,$$

$$\frac{a}{b}\le c_0.$$

$$\underset{X\in C_0^1(U),|X|\le 1}{\sup }\left|\int \mathrm{div}Xd{\mu }_V\right|\le c.$$

$$\int \mathrm{div}X=-\int H\cdot X.$$

$$\frac{{\int }_{\{u>0\}}|\nabla u{|}^{p-2}\left(|\nabla \varphi {|}^2+(p-2)\frac{(\nabla u\cdot \nabla \varphi {)}^2}{|\nabla u{|}^2}\right)-{\int }_{\partial \{u>0\}}\kappa {\varphi }^2}{{\int }_{\{u>0\}}|\nabla u{|}^{p-2}{\varphi }^2}\ge -C_0$$

$${\int }_{\partial \{u>0\}}\kappa {\varphi }^2\le {\int }_{\{u>0\}}|\nabla u{|}^{p-2}\left(|\nabla \varphi {|}^2+(p-2)\frac{(\nabla u\cdot \nabla \varphi {)}^2}{|\nabla u{|}^2}\right)\forall \varphi \in C_0^{0,1}({\mathbb{R}}^2\setminus \{0\}).$$

$$\xi_{k}(x)=\left\{\begin{array}[]{ll}0&\textrm{in}\ B_{r_{k+1}},\\ 1-\frac{1}{\log 2}\log\frac{r_{k}}{|x|}&\textrm{in}\ B_{r_{k}}\setminus B_{r_{k+1}},\\ 1&\textrm{in}\ B_{1}\setminus B_{r_{k}},\end{array}\right.$$

$${\int }_{\partial \{u>0\}}\kappa {\varphi }_k^2$$

$${\int }_{\{u>0\}}|\nabla {\varphi }_k{|}^2$$

$${\int }_{\partial \{u>0\}}\kappa {\varphi }_k^2\le \frac{2\pi }{\log 2}+C(\eta )+C_0|B_1|.$$

$$2\pi N\le \frac{2\pi }{\log 2}+C(\eta )+C_0|B_1|,$$

$$u_{\epsilon }:=\max \{u-\epsilon \zeta ,0\},$$

Full text

[TABLE]

where $\mathscr{L}u=\mathrm{div}(g(\nabla u)\nabla u)$ is a quasilinear elliptic operator and $g(\xi)$ is a given function of $\xi$ satisfying some structural conditions. We prove that the free boundary $\partial\{{u}>0\}$ is continuously differentiable in ${\mathbb{R}}^{2}$, provided that $\{{u}>0\}$ has locally finite connectivity. Moreover, we show that the free boundaries of weak solutions with finite Morse index must have finite connectivity. The weak solutions are locally Lipschitz continuous and non-degenerate stationary points of the Alt-Caffarelli type functional $J[u]=\int_{\Omega}F(\nabla u)+Q^{2}\chi_{\{u>0\}}$.

The full regularity of the free boundary is not fully understood even for the minimizers of $J[u]$ in the simplest case $g(\xi)=|\xi|^{p-2},p>1$, partly because the methods from the classical case $p=2$ cannot be generalized to the full range of $p$. Our method, however, is very geometric and works even for the stationary points of the functional $J[u]$ for a large class of nonlinearities $F$.

2000 Mathematics Subject Classification. Primary 35R35, 35J60.

Keywords: Free boundary regularity, Morse index, global solutions, unique continuation.

1. Introduction

In this paper we study the weak solutions of the free boundary problem

[TABLE]

where $\mathscr{L}u=\mathrm{div}(\rho(\nabla u)\nabla u)$ is a quasilinear elliptic operator, $B_{r}(x_{0})=\{x\in{\mathbb{R}}^{2}:|x-x_{0}|<r\}$, $\rho(\xi)$ is a given function of $\xi\in{\mathbb{R}}^{2}$ subject to some standard structural conditions, $\ell$ is a constant. The solutions of (1.1) can be seen as stationary points of the functional

[TABLE]

where $W^{1,F}(\Omega)$ is the Orlicz-Sobolev space of function defined on a Lipschitz domain $\Omega\subset{\mathbb{R}}^{2}$, $u_{0}\in W^{1,F}(\Omega)$ is a given boundary condition, $\chi_{D}$ is the characterisitic function of a set $D$, and $\ell$ is determined from the implicit relation

[TABLE]

where $Q>0$ determines the Bernoulli condition along the free boundary. One can write (1.1) in a more concise form

[TABLE]

where $\partial_{\rm red}\{{u}>0\}$ is the reduced boundary of the set $\{{u}>0\}$, $\mathcal{H}^{s}$ is the $s-$dimensional Hausdorff measure, and $\lambda_{0}=F^{\prime}(\ell)$. Note that the latter relation can be used to recover the free boundary condition $|\nabla u|=\ell$ from the equation $\mathscr{L}u=\lambda_{0}\mathcal{H}^{1}\with\partial_{\rm red}\{{u}>0\}$. We give the precise definition below, which is valid in ${\mathbb{R}}^{n},n\geq 2.$

Definition 1.1**.**

A function $u$ is said to be a weak solution of $\mathscr{L}u=\lambda_{0}\mathcal{H}^{n-1}\with\partial_{\rm red}\{{u}>0\}$ in a domain $\Omega\subset{\mathbb{R}}^{n}$ if the following is satisfied:

$u\in W^{1,F}(\Omega)$* is continuous and non-negative in $\Omega$ and $\mathscr{L}u=0$ in $\{u>0\}$,*

• 2)

for every bounded subdomain $D\Subset\Omega$ there are constants $0<c_{\min}\leq C_{\max}<\infty$ depending only on $n,\ell,\rho,\operatorname{dist}(\partial D,\Omega)$ such that for every ball $B_{r}(x)\subset D$ centered at free boundary point $x\in\partial\{u>0\}$ the following inequalities hold

[TABLE]

• 3)

$\{u>0\}$* is a set of locally finite perimeter and $\mathscr{L}u=\lambda_{0}\mathcal{H}^{n-1}\with\partial_{\rm{red}}\{u>0\}$ in the following sense: for test function $\zeta\in C_{0}^{\infty}(\Omega)$ the equality*

[TABLE]

holds. Here $\partial_{\rm red}\{u>0\}$ is the reduced boundary of $\{u>0\}$, see 4.5.5 [Federer] for definition.

Remark 1.1**.**

In [Wolan-adv] a ”flatness implies regularity“ type result is proven for weak solutions. If $x_{0}\in\partial_{\rm red}\{u>0\}$ then the free boundary near $x_{0}$ is a smooth surface. Hence, the free boundary condition $|\nabla u|=\ell$ is satisfied in the classical sense. Furthermore, the weak solutions enjoy the following properties:

• $\bf 1^{\circ}$

$\partial\{{u}>0\}$* is a set of locally finite perimeter and $\partial_{\rm red}\{u>0\}$ is open relative to $\partial\{u>0\}$,*

• $\bf 2^{\circ}$

$\partial_{\rm red}\{u>0\}$* is smooth, and $\mathcal{H}^{n-1}(\partial\{u>0\}\setminus\partial_{\rm red}\{u>0\})=0$,*

• $\bf 3^{\circ}$

the gradient is upper-semicontinuous, i.e.

[TABLE]

Our first result states that the finitely connected free boundaries of weak solutions are smooth in ${\mathbb{R}}^{2}$. To elucidate our method we first choose to formulate the result for $\mathscr{L}u=\mathrm{div}(|\nabla u|^{p-2}\nabla u)$.

Theorem 1.2**.**

Let $\mathscr{L}=\Delta_{p}$ be the $p-$laplacian, i.e. $\rho(\xi)=|\xi|^{p-2},F(\xi)=|\xi|^{p},1<p<\infty$ and $u$ be a weak solution of (1.1) in $B_{1}$, the unit ball centered at the origin, in the sense of Definition 1.1. Suppose that there is a constant $\theta>0$ such that

[TABLE]

If $\partial\{{u}>0\}$ has finite connectivity then it is a finite collection of continuously differentiable curves in $B_{1}$.

One can show that the finite Morse index free boundaries with nonnegative generalized curvature must be of finite connectivity.

Theorem 1.3**.**

Let $u$ be as in Theorem 1.2. If $p\geq 2$ and the curvature measure $\kappa$ of $\partial\{{u}>0\}$ exists and it is nonnegative then the finite Morse index solutions have locally finitely connected free boundaries.

The Morse index of some $x_{0}\in\partial\{{u}>0\}$ is defined in standard way, that is, it is the number $m$ of negative eigenvalues $\theta_{1},\dots,\theta_{m}$ of the problem $-\mathrm{div}(A\nabla\phi)-\kappa\phi\mathcal{H}^{1}\with\partial_{\rm red}\{{u}>0\}=\theta|\nabla u|^{p-2}\phi$ in ${\mathbb{R}}^{2}\setminus\{x_{0}\}$ where $A_{ij}=|\nabla u|^{p-2}(\delta_{ij}+(p-2)u_{i}u_{j}|\nabla u|^{-2})$. This eigenvalue problem arises naturally when one computes the second variation of the energy [CJK], [K18]. The zero Morse index solutions are the minimizers, see Section 3 for precise definitions.

2. The curvature of free boundary

The proof of Theorem 1.2 is a combination of two lemmas to follow. In the first one we show that any component of $\partial\{{u}>0\}$ with positive $\mathcal{H}^{1}$ measure must be smooth.

Lemma 2.1**.**

Let $u$ be as in Theorem 1.2 and $\gamma\subset\partial\{{u}>0\}$ a component of free boundary such that $\mathcal{H}^{1}(\gamma)>0$. Then $\gamma$ is convex and smooth.

**Proof. **We employ a compactness argument to show that at every point of $\gamma$ the blow-up limit $u_{0}$ exists and $u_{0}$ is a weak solution thanks to condition (1.4). Consider $u_{k}(x)=\frac{u(x_{0}+r_{k}x)}{r_{k}}$ for some positive sequence $r_{k}\downarrow 0$ with $x_{0}\in\gamma\subset\partial\{u>0\}$. From the Lipschitz continuity of $u$ [DP-2D, Wolan], it follows that $\{u_{k}\}$ is locally uniformly Lipschitz. By a customary compactness argument there exists a subsequence, still denoted $\{u_{k}\}$, converging to a limit $u_{0}\in W^{1,\infty}_{\rm loc}({\mathbb{R}}^{2})$ such that

[TABLE]

For the proofs of (2.1)-(2.3) we refer the reader to [DP-2D], [Wolan-adv].

Let $u_{0}$ be a blow-up of $u$ at $x_{0}\in\gamma$, then by Proposition 5.1 (see Appendix) $u_{0}$ is a weak solution. Therefore from Remark 1.1 we have $|\nabla u_{0}|=\ell$ on $\partial_{\rm red}\{{u_{0}}>0\}$ and

[TABLE]

Let $S\subset\partial_{\rm red}\{{u_{0}}>0\}$ be a smooth connected curve and $S^{\prime}\subset\{{u_{0}}>0\}$ a smooth perturbation of $S$ such that $S$ and $S^{\prime}$ have the same endpoints. Consider the domain $D\subset\{{u_{0}}>0\}$ bounded by $S$ and $S^{\prime}$, i.e. $\partial D=S\cup S^{\prime}$. We have

[TABLE]

Utilizing the estimate $|\nabla u_{0}(x)|\leq\ell,x\in{\mathbb{R}}^{2}$ (see Remark 1.1 ${\bf 3^{\circ}}$) we infer

[TABLE]

Since $S$ is smooth we can locally paramatrize it as $x_{2}=h(x_{1}),x\in(-\delta,\delta)$ for suitable choice of coordinate axis $x_{1},x_{2}.$ Suppose that $S^{\prime}$ is given by $x_{2}=h(x_{1}-t\psi(x_{1}))$, where $t>0$ is small and $0\leq\psi\in C_{0}^{\infty}(-\delta,\delta)$. Then from (2.5) we have

[TABLE]

Therefore $\frac{d}{dx_{1}}\left(\frac{h^{\prime}}{\sqrt{1+(h^{\prime})^{2}}}\right)\geq 0$ on $(-\delta,\delta)$. Consequently, the outer curvature $\kappa(S)\geq 0$, i.e. $S$ is a convex graph in $x_{2}$ direction. Since $u_{0}$ is a weak solution then it follows that $\partial\{{u_{0}}>0\}\cap B_{1}$ is rectifiable and therefore $\partial\{{u_{0}}>0\}\cap B_{1}=\gamma_{0}\cup\left(\cup_{k=1}^{\infty}\gamma_{k}\right)$ such that $\gamma_{k},k\geq 1$ are differentiable curves and $\mathcal{H}^{1}(\gamma_{0})=0$. Moreover, by (2) every $\gamma_{k},k\geq 1$ is convex. For some fixed $k_{0}$ let $y_{0}\in\partial\gamma_{k_{0}}$, the relative boundary of $\gamma_{k_{0}}$. Observe that $\gamma_{k_{0}}$ is convex so there is one sided sub-differential at $y_{0}$ (from the regular side of $\gamma_{k_{0}}$). If we blow-up $u_{0}$ at $y_{0}$, then $u_{00}$ is again a weak solution thanks to Proposition 5.1. Moreover, at $0\in\partial\{{u_{00}}>0\}$, the free boundary contains a line on which $|\nabla u_{00}|=\ell$. Without loss of generality (because $\rho(\xi)$ is rotation invariant) we may assume that the positive semiaxis $x_{1}>0$ is a subset of the free boundary $\partial\{{u_{00}}>0\}$ and $\mathscr{L}u_{00}=0$ in $\{{u_{00}}>0\}$. Continuing $u_{00}$ linearly across the positive semiaxis $x_{1}>0$ and letting

[TABLE]

we see that $\mathscr{L}\widetilde{u}_{00}=0$ in some neighborhood of $x_{1}=0$. Applying the unique continuation property of $\mathscr{L}$ [GM] we infer that $u_{00}(x_{1},x_{2})=\ell x_{1}^{+}$. This implies that $u_{0}$ is flat at $y_{0}$ and the relative boundary of the convex arc $\gamma_{k_{0}}$ is empty. In other words, $\partial\{{u_{0}}>0\}$ is a smooth, convex, and complete curve in ${\mathbb{R}}^{2}$. This yields that $u$ is flat at $x_{0}$. Indeed, since $\partial\{{u_{0}}>0\}$ is smooth then at $0\in\partial\{{u_{0}}>0\}$ we can take $\rho>0$, small, such that

[TABLE]

where $e$ is the normal of $\partial\{{u_{0}}>0\}$ at [math] and $\bar{\sigma}_{0}$ is the critical flatness constant, see Theorem 9.3 [Wolan-adv]. In other words, $u_{0}$ belongs to the flatness class $F\left(\frac{\bar{\sigma}_{0}}{2},1;\infty\right)$ in $e$ direction. Choose a sequence $r_{k}\to 0$ as above such that $u_{k}(x)=\frac{u(x_{0}+r_{k}x)}{r_{k}}\to u_{0}(x)$ and (2.1)-(2.3) hold. In particular from (2.2) it follows that

[TABLE]

or $u_{k}\in F\left(\frac{3\bar{\sigma}_{0}}{4},1;\infty\right)$ for sufficiently large $k$. Applying Theorem 9.3 [Wolan-adv] we get that $B_{\rho/4}\cap\partial\{{u_{k}}>0\}$ is $C^{1,\alpha}$ surface $\alpha>0$ in $e$ direction. Consequently, pulling back to $u$ and using flatness implies regularity result (Theorem 9.4 [Wolan-adv]) we infer that $\partial\{{u}>0\}$ is differentiable at $x_{0}$ and hence smooth. ∎

Recall that if $K\subset{\mathbb{R}}^{2}$ is bounded convex set with non-empty interior then there is an ellipse $E\supset K$ of minimal area and center $x_{0}\in K$ so that

[TABLE]

see [deGuzman] Lemma 2.2 page 139.

Corollary 2.2**.**

Let $u$ be as in Theorem 1.2. Let $\gamma$ be a closed convex component of $\partial\{{u}>0\}$ and $a\geq b$ are the semiaxis of John’s ellipse of the convex hull of $\gamma$. Then there is universal $c_{0}$ such that

[TABLE]

**Proof. **Suppose the claim fails, then there are convex closed curves $\gamma_{k}\subset\partial\{{u}>0\}$ such that the semiaxis $a_{k}\geq b_{k}$ of John’s ellipse $E_{k}$ satisfy $\frac{a_{k}}{b_{k}}\to\infty.$ Let $z_{k}$ be a free boundary point intersecting the major semiaxis of $E_{k}$. Consider $u_{k}=\frac{u(z_{k}+a_{k}x)}{a_{k}}$. Then using a customary compactness argument (as in the proof of Lamma 2.1) we can extract a subsequence such that $u_{k}$ converges to a weak solution $u_{0}$ for which (1.4) fails. In view of Proposition 5.1 this is a contradiction. ∎

3. Finite Morse index solutions

In this section we prove Theorem 1.3. We show that the finite index free boundary of $u$ is ”non-thinning”, i.e. it is not possible to have $x_{0}\in\partial\{{u}>0\}$ and disjoint components $\gamma_{k}\subset\partial\{{u}>0\}$ such that $x_{0}\not\in\gamma_{k},k=1,2,\dots$ but for some sequence $x_{k}\in\gamma_{k}$ we have $x_{k}\to x_{0}$. Obviously, if $\partial\{{u}>0\}$ is thinning then we can assume that each $\gamma_{k}$ must be a closed convex curve. Indeed, if $u_{0}$ is a blow up of $u$ at $x_{0}$ then $\partial\{{u_{0}}>0\}$ is thinning too and by lemma 2.1 each nontrivial bounded component of the free boundary is a closed convex smooth curve.

Recall that by Theorem 6.2 [ACF-Q] $\Gamma$ is rectifiable. Then we define the rectifiable $1$ varifold $V=v(\Gamma,\theta)$ (with multiplicity $\theta$) as the equivalence class of all pairs $(\widetilde{\Gamma},\widetilde{\theta})$ such that $\widetilde{\Gamma}$ is $\mathcal{H}^{1}$ rectifiable, $\mathcal{H}^{1}(\Gamma\triangle\widetilde{\Gamma})=0$ and $\theta=\widetilde{\theta}$ a.e. on $\Gamma\cap\widetilde{\Gamma}$ [Simon] page 77. We say that $V$ has bounded first variation if there is a constant $c>0$ such that

[TABLE]

By the Riesz represenation theorem there is a vector measure $H$ such that

[TABLE]

$\kappa:=|H|$ is called the curvature measure of the varifold $V$. If $\Gamma$ is smooth then $H$ coincides with the curvature vector.

Definition 3.1**.**

Let $u$ be a weak solution and the curvature measure of $\partial\{{u}>0\}$ is locally finite. We say that $u$ has finite Morse index at $x_{0}\in\partial\{{u}>0\}$ if there is a constant $C_{0}>0$ such that

[TABLE]

whenever $\phi\in C_{0}^{0,1}({\mathbb{R}}^{2}\setminus\{x_{0}\}).$ Here $\kappa$ is the curvature measure of the free boundary. If $x_{0}$ has finite index then we write $\mathrm{Index}(x_{0})<\infty$.

The second variation of the energy $J[u]$ is computed in [K18], where the following stability inequality for the minimizers had been derived

[TABLE]

Thus finite Morse index means that there are finitely many negative eigenvalues if one minimizes the ratio in (3.1).

Lemma 3.1**.**

Let $u$ be as in Theorem 1.3. Suppose $0\in\partial\{{u}>0\}$ and $\mathrm{Index}(0)<\infty$. Then $\partial\{{u}>0\}$ has finite connectivity near $0.$

**Proof. **If there is a free boundary component joining [math] with some nonzero point then the argument in the proof of Lemma 2.1 shows that $\partial\{{u}>0\}$ must be a smooth curve near $u$. Thus we assume that $\partial\{{u}>0\}$ is thinning near [math]. Let $r_{k}=2^{-k}$ and define

[TABLE]

and, set $\phi_{k}=\eta\xi_{k}$ where $\eta\in C_{0}^{\infty}(B_{1})$ such that $0\leq\eta\leq 1$ and $\eta=1$ in $B_{\frac{1}{2}}$. Utilizing (3.1) with $\phi=\phi_{k}$ gives

[TABLE]

where the last inequality follows from (2.4). On the other hand we get

[TABLE]

Hence, returning to (3) we get

[TABLE]

If $\gamma_{1},\dots,\gamma_{N}$ are closed convex components of $\partial\{{u}>0\}\cap B_{\frac{1}{2}}$ then from (3.5) (with sufficiently large $k$) we get that

[TABLE]

where we used $\int_{\gamma_{i}}\kappa=2\pi$. Thus $N$ is finite and $\{{u}>0\}$ has finite connectivity at [math]. ∎

Remark 3.2**.**

There are examples of periodic weak solutions in ${\mathbb{R}}^{2}$ as a solution to a stationary problem in hydrodynamics [Baker], [Cro]. In [T] Traizet gave the full classification of smooth free boundaries with finite connectivity in ${\mathbb{R}}^{2}$. In this respect Theorem 12 in [T] says that if $\partial\{{u}>0\}$ has infinite connectivity then it cannot have finite Morse index.

Remark 3.3**.**

It is well known that a minimizer of $J_{p}[u]=\int_{B_{1}}|\nabla u|^{p}+\lambda\chi_{\{u>0\}}$ is also a weak solution. For $\varepsilon>0$ small, let

[TABLE]

where $\zeta\in C_{0}^{0,1}(B_{\rho}(x_{0}))$, defined as

[TABLE]

Then in ${\mathbb{R}}^{2}$ the comparison $J_{p}[u]\leq J_{p}[u_{\varepsilon}]$ gives that every blow-up of $u$ must have constant gradient provided $p>2$, see [AC], [ACF-Q], [DP-2D]. Clearly, this argument cannot be used if $u$ is merely a stationary point. Moreover, even for the minimzers, it does not imply that the free boundary is continuously differentiable for the range $1<p<2$, cf [DP-2D].

In closing this section we construct a sequence of weak solutions such that the measure theoretic boundary of its limit is empty, cf. [AC] 5.8. This example shows that the condition (1.4) is necessary. Let us define

[TABLE]

One can check that $u$ is a weak solution for every $\varepsilon>0.$ However, for $\varepsilon=0$ this is not true. In this case $\Delta_{p}u=2\mathcal{H}^{n-1}\with\partial\{{u}>0\}$ and $\mathcal{H}^{n-1}(\partial\{u>0\}\setminus\partial_{\rm{red}}\{u>0\})>0$ since $\partial_{\rm{red}}\{u>0\}=\emptyset$, in other words the normal derivative $\partial_{\nu}u$ cannot be reconstructed from the free boundary data$\ell=2$.

4. Generalizations

One can impose various assumptions on $\rho$ to guarantee that the elliptic operator has nice properties. We formulate them in the following three hypotheses:

• (H1)

$\mathscr{L}$ is a quasilinear elliptic operator such that the strong maximum principle, interior $C^{1,\alpha}$ regularity theory, Harnack inequality for non-negative solutions are valid for the weak solutions of $\mathscr{L}u=0$. Under these conditions it follows that if $u$ is a weak solution of (1.1) then the gradient is upper semicontinuous, i.e.

[TABLE]

• (H2)

The unique continuation for the weak solutions of $\mathscr{L}u=0$ is valid, i.e. if $\mathscr{L}u=0$ in $\Omega$ and there is subdomain $\Omega^{\prime}\subset\Omega$ such that $u$ is affine on $\Omega^{\prime}$ then $u$ is affine in $\Omega.$

• (H3)

The class of weak solutions is closed with respect to blow-up.

If $\bf(H1)$-$\bf(H3)$ are valid then one can generalize the variational theory from [AC], [ACF-Q] for a larger class of equations

[TABLE]

where $g(t)=G^{\prime}(t)$ and $G$ is a convex function satisfying $0<c<\frac{G^{\prime}(t)}{t}<C<\infty$ with implicitly given free boundary conditions

[TABLE]

where $\Psi$ is determined by $g$, see [Wolan-adv].

Some examples are as follows:

• •

The classical Alt-Caffarelli functional $J_{\rm AC}[u]=\int_{\Omega}|\nabla u|^{2}+Q^{2}\chi_{\{u>0\}}$ [AC] where $Q(x)>0,x\in\Omega$ is a Hölder continuous function bounded away from zero and infinity.

• •

The nonlinear version of $J_{\rm AC}$

[TABLE]

with $c_{0}\leq F^{\prime}(t)\leq C_{0},0\leq F^{\prime\prime}(t)\leq C_{0}(1+t)^{-1}$ [ACF-Q]. The weak solutions solve $\mathscr{L}u=\lambda_{0}\mathcal{H}^{1}\with\partial_{\rm red}\{{u}>0\}$ with implicitly defined free boundary condition

[TABLE]

• •

The non-radially symmetric version of $J_{\rm ACF}$, namely

[TABLE]

under the assumtion that $p\cdot\nabla f(p)-f(p)$ is convex [Weiss-95].

• •

The analogue of minimiziation problem for the functional $J_{\rm ACF}$ in the Orlicz-Sobolev spaces corresponding to the energy $\int_{\Omega}G(|\nabla u|)+Q^{2}\chi_{\{u>0\}}$ where

[TABLE]

and the free boundary condition is $|\nabla u|=\ell$ where $\ell$ is determined from the implicit relation $G^{\prime}(\ell)\ell-G(\ell)=Q^{2}$ [Wolan-adv]. The weak equation is $\mathscr{L}u=\lambda_{0}\mathcal{H}^{1}\with\partial_{\rm red}\{{u}>0\}$ and $\lambda_{0}=g(\ell)$.

• •

The $p(x)$-Laplacian model and the variable growth functional

[TABLE]

modelling the stationary flow of electrorheological fluids. In this case the free boundary condition is $\ell:=|\nabla u|$ and the differential operator $\mathscr{L}u:=\Delta_{p(x)}u=\mathrm{div}(|\nabla u|^{p(x)-2}\nabla u)$ with $\lambda_{0}=\ell^{p(x)-1}$, for constant case $p(x)=p$ see [DP-2D].

In order to formulate the general result it is convenient to introduce the following classes $\mathcal{P}_{r}(x_{0},M)$ of weak solutions:

Definition 4.1**.**

We say that $u\in{\mathcal{P}}_{r}(x_{0},M)$ if

• (a)

$u\in C^{0,1}(B_{r}(x_{0}))$* and $\sup_{r}|\nabla u|\leq M$,*

• (b)

$u\geq 0$* and $x_{0}\in\partial\{{u}>0\}$,*

• (c)

$u$* is a weak solution in $B_{r}(x_{0})$.*

We say that $u\in{\mathcal{P}}_{r}(x_{0},M,\theta)$ if $u\in{\mathcal{P}}_{r}(x_{0},M)$ and

[TABLE]

for any $B_{\rho}(x)\subset B_{r}(x_{0}),x\in\partial\{{u}>0\}$.

Note that if $u\in P_{r}(x_{0},M)$ and $u_{s}(x)=\frac{u(x_{0}+sx)}{s}$ then $u_{s}\in{\mathcal{P}}_{r/s}(0,M)$. Moreover, in view of Proposition 5.1 (see Appendix) if $u\in P_{r}(x_{0},M,\theta)$ and $u_{s_{k}}\to u_{0}$ locally uniformly (for some $0<s_{k}\downarrow 0$) then $u_{0}\in{\mathcal{P}}_{\infty}(0,M,\theta)$. Therefore $\bf(H3)$ is valid for ${\mathcal{P}}_{r}(x_{0},M,\theta)$.

From the proof of Theorem 1.2 we have the following generalization.

Theorem 4.1**.**

Let $u\in{\mathcal{P}}_{1}(x_{0},M,\theta)$ such that the hypotheses (H1)-(H3) are satisfied, then near $x_{0}$ the free boundary is a continuously differentiable curve provided that $\{{u}>0\}$ has finite connectivity.

5. Appendix

In this section we prove that ${\mathcal{P}}_{r}(x_{0},M,\theta)$ is closed with respect to the blow-up procedure. We choose to state the result in ${\mathbb{R}}^{n}$ and adapt the proof from [Weiss].

Proposition 5.1**.**

Let $u$ be a weak solution of $\mathscr{L}u=\lambda_{0}\mathcal{H}^{n-1}\with\partial\{u>0\}$ in the sense of Definition 1.1. If $u\in{\mathcal{P}}_{r}(x_{0},M,\theta)$ such that the blow-up sequence $u_{k}(x)=\frac{u(x_{0}+\rho_{k}x)}{\rho_{k}}$ converges locally uniformly to $u_{0}$, then $u_{0}\in{\mathcal{P}}_{R}(0,M,\theta)$ for any $R>0$.

Proof. Step 1) Let $u_{0}$ be a blow-up limit, i.e. let $u_{k}(x)=\frac{u(x_{0}+\rho_{k}x)}{\rho_{k}}$, $x_{0}\in\partial\{u>0\}$ for some sequence $\rho_{k}\searrow 0,k\rightarrow\infty$. Then by a customary compactness argument [Wolan-adv] Lemma 7.1 and Remark 8.2 we can extract a subsequence, still denoted by $\rho_{k}$, such that $u_{k}\to u$ locally uniformly. More precisely, we have that

[TABLE]

is a weak solution. Consequently, it follows that properties 1) and 2) in Definition 1.1 for $u_{0}$ hold true. Furthermore,

[TABLE]

To show that $u_{0}$ is weak solution it remains to verify the equation 3) in the Definition 1.1. We need to show two things: $\mathcal{H}^{n-1}(\partial\{u_{0}>0\}\setminus\partial_{\rm red}\{u_{0}>0\})=0$ and the smoothness of $\partial_{\rm red}\{u_{0}>0\}$ stated in the Remark 1.1 $\bf 1^{\circ}$-$\bf 3^{\circ}$.

Step 2) Next, we prove that $\{u_{0}>0\}$ is of finite perimeter. Take $\zeta(x)=\max(0,\min(1,\frac{1}{\varepsilon}(R-|x|)))$ in 3) of Definition 1.1 to conclude, after sending $\varepsilon$ to zero, that for a.e. $R>0$

[TABLE]

From here the claim follows from the semi-continuity of the perimeter.

Since the current boundary $T=\partial({\mathbb{R}}^{n}\with\{u_{0}>0\}\cap B_{R}(0))$ is representable by integration, $\|T\|=\int_{B_{R}(0)}|D\chi_{\{u_{0}>0\}}|$, we get from 4.5.6. (3) [Federer] $\partial\{u_{0}>0\}\setminus\partial_{\rm red}\{u_{0}>0\}=K_{0}\cup K_{+}$ where $\mathcal{H}^{n-1}(K_{+})=0$ and for $x_{1}\in K_{0}$, $r^{1-n}\mathcal{H}^{n-1}(\partial_{\rm red}\{u_{0}>0\}\cap B_{r}(x_{1}))\rightarrow 0$ as $r\rightarrow 0$.

Let us show that $K_{0}=\emptyset$. For $u_{0k}(x)=\frac{u_{0}(x_{1}+r_{k}x)}{r_{k}}$, by compactness argument as above, we have $u_{0k}\rightarrow u_{1}$ for another function $u_{1}$ and

[TABLE]

On the other hand

[TABLE]

Hence we infer that $\chi_{\{u_{1}>0\}}$ is a function of bounded variation which is constant a.e. in $B_{R}$. The positive Lebesgue density property of $\{u_{1}=0\}$, translated to $u_{1}$ through compactness as in step 1), and strong maximum principle for the solutions of $\mathscr{L}u=0$ (see Hypothesis (H1)) demands $u_{1}$ to be zero. This is in contradiction with 2) in Definition 1.1 since by compactness as in step 1) the non-degeneracy translates to $u_{1}$. Thus $K_{0}=\emptyset$ and we obtain $\mathcal{H}^{n-1}(\partial\{u_{0}>0\}\setminus\partial_{\rm red}\{u_{0}>0\})=0$.

To show that $\partial_{\rm red}\{u_{0}>0\}$ is smooth, we notice that if $z_{0}\in\partial_{\rm red}\{u_{0}>0\}$ and $\nu(z_{0})$ is the normal at $z_{0}$ in the sense of 4.5.5. [Federer], then by the uniform Lebesgue density of $\{u_{0}=0\}$ and nondegeneracy of $u_{0}$ from 2) of Definition 1.1 we have $u_{0}\in F(\frac{\sigma}{2},1;\infty)$ in $B_{2\rho}(z_{0})$ in direction $e=\nu(z_{0})$ for $\sigma\leq\sigma_{0}$ and $\rho\leq\tau_{0}\sigma^{\frac{2}{\beta}}$. Here $F$ is the flatness class defined as in [Wolan-adv] definition 9.1.

At this point we don’t know if $u_{0}$ is a weak solution so we cannot immediately apply the ”flatness implies regularity“ to $u_{0}$. However from (5.2) we conclude that $u_{k}$, which is a weak solution, is in $F(\sigma,1;\infty)$ in $B_{\rho}(z_{0})$ in the direction of $e$ for sufficiently large $k$ (this is because $z_{0}\in\partial_{\rm red}\{{u_{0}}>0\}$ and $e$ is the normal at $z_{0}$ in measure theoretic sense). Therefore, the surfaces $\partial\{{u_{k}}>0\}$ are all graphs of $C^{1,\alpha}$ functions in direction $e$ with uniform bounds so that we get the same property for $\partial\{{u_{0}}>0\}$ Thus $\partial\{u_{k}>0\}$ are $C^{3}$ smooth in $B_{\frac{\rho}{4}}(z_{0})$ in the direction of $e$ and this translates to $\partial\{u_{0}>0\}$ in $B_{\frac{\rho}{4}}(z_{0})$.

Step 3) Now we can finally show that $u_{0}$ satisfies the equation in 3) of Definition 1.1. Take a compactly supported smooth function $\zeta$ and fix a $\delta>0$ small. Let $\mathscr{F}_{1}$ be a finite subcovering of $\operatorname{supp}\zeta\cap(\partial\{u_{0}>0\}\setminus\partial_{\rm red}\{u_{0}>0\})$ by balls $B_{t_{i}}(y_{i})$ such that $\displaystyle\sum_{i=1}^{N^{\prime}(\delta)}t_{i}^{n-1}<\delta$, see step 2). Then using partial integration we get

[TABLE]

To get the last line we used the definition of $\lambda_{0}=g(\ell)$ and that by step 2) $\partial_{\rm red}\{{u_{0}}>0\}$ is smooth, hence, the free boundary condition $|\nabla u_{0}|=\ell$ holds in the classical sense. Sending $\delta\rightarrow 0$ we conclude that $u_{0}$ is a weak solution. ∎