A free boundary problem on three-dimensional cones

TL;DR

This paper investigates a free boundary problem on three-dimensional cones, identifying conditions under which the free boundary passes through the vertex, and establishes the critical dimension for this behavior.

Contribution

It determines the critical dimension for free boundary passage through the vertex of cones, linking geometric analysis with free boundary problem behavior.

Findings

For large c, the free boundary avoids the vertex.

For small positive c, the free boundary passes through the vertex.

Identifies 3 as the critical dimension for this phenomenon.

Abstract

We consider a free boundary problem on cones depending on a parameter c and study when the free boundary is allowed to pass through the vertex of the cone. We show that when the cone is three-dimensional and c is large enough, the free boundary avoids the vertex. We also show that when c is small enough but still positive, the free boundary is allowed to pass through the vertex. This establishes 3 as the critical dimension for which the free boundary may pass through the vertex of a right circular cone. In view of the well-known connection between area-minimizing surfaces and the free boundary problem under consideration, our result is analogous to a result of Morgan that classifies when an area-minimizing surface on a cone passes through the vertex.

Equations

$$\Delta u$$

$$|\nabla u|$$

$$\begin{aligned} \partial_{j}(a^{ij}(x)u_{i})&=0&\text{ in }&\{u>0\}\\ a^{ij}(x)u_{i}u_{j}&=1&\text{ on }&\partial\{u>0\}.\end{aligned}\$$

$${\int }_{\Omega }{a}^{ij}(x)v_i{v}_j+Q(x){\chi }_{\{v>0\}}.$$

$$x_{n+1}=c\sqrt{\sum _{i=1}^n{x}_i^2}$$

$$k\ge 3\text{ and }{\delta }^2\ge \frac{4(k-1)}{k^2},$$

$$\mathcal{C}:=\{(y_1,y_2,y_3,y_4)\in {\mathbb{R}}^4:y_4=c\sqrt{y_1^2+y_2^2+y_3^2}\}.$$

$$J(v,\Omega ):={\int }_{\Omega }|{\nabla }_{c}v{|}^2+{\chi }_{\{v>0\}},$$

$$\{\begin{array}{l}u\ge 0\\ u\text{ is continuous in }\Omega \\ {\Delta }_{c}u=0\text{ in }\{u>0\}\\ \partial \{u>0\}\setminus \{0\}\text{ is locally smooth}\\ |{\nabla }_{c}u|=1\text{ on }\partial \{u>0\}\setminus \{0\},\end{array}$$

$$y_1=r\cos \theta \sin \varphi ,y_2=r\sin \theta \sin \varphi ,y_3=r\cos \varphi ,y_4=cr.$$

$$\left(\begin{array}[]{ccc}r^{-2}\sin^{-2}\phi&0&0\\ 0&r^{-2}&0\\ 0&0&(1+c^{2})^{-1}\end{array}\right),$$

$${\int }_{\Omega }\sqrt{g}{g}^{ij}{u}_i{u}_j+\sqrt{g}{\chi }_{\{u>0\}}.$$

$$\{\begin{array}{ll}\frac{1}{\sqrt{g}}{\partial }_j(\sqrt{g}{g}^{ij}{u}_i)& =0\text{ in }\{u>0\}\\ \sqrt{g}{g}^{ij}{u}_i{u}_j& =\sqrt{g}\text{ on }\partial \{u>0\}\setminus \{0\},\end{array}$$

$$\frac{1}{1+c^2}\left(u_{rr}+\frac{2}{r}{u}_r\right)+\frac{1}{r^2}\left(\frac{u_{\theta \theta }}{\sin \varphi }+\frac{\cos \varphi }{\sin \varphi }{u}_{\varphi }+u_{\varphi \varphi }\right)=0$$

$$y_1=x_1,y_2=x_2,y_3=x_3,y_4=c\sqrt{x_1^2+x_2^2+x_3^2}.$$

$$\frac{1}{(1+c^{2})r^{2}}\left(\begin{array}[]{ccc}r^{2}+c^{2}(x_{2}^{2}+x_{3}^{2})&-c^{2}x_{1}x_{2}&-c^{2}x_{1}x_{3}\\ -c^{2}x_{1}x_{2}&r^{2}+c^{2}(x_{1}^{2}+x_{3}^{2})&-c^{2}x_{2}x_{3}\\ -c^{2}x_{1}x_{3}&-c^{2}x_{2}x_{3}&r^{2}+c^{2}(x_{1}^{2}+x_{2}^{2})\end{array}\right).$$

$$C_{1}r\le \underset{{\mathcal{C}}_r}{\sup }u\le C_{2}r.$$

$$W(r,u):=\frac{1}{r^n}{\int }_{{\mathcal{C}}_r}|{\nabla }_{c}u{|}^2+{\chi }_{\{u>0\}}-\frac{1}{r^{n+1}}{\int }_{\partial {\mathcal{C}}_r}{u}^2$$

$${\int }_{\partial \Gamma }H{F}^{2}d\sigma \le {\int }_{\Gamma }{g}^{ij}{F}_i{F}_j\text{ for every }F\in \mathcal{F}$$

$${\Omega }_{ϵ}=\{x\in \Omega :u(x)>ϵF(x)\}.$$

$${\int }_{B_R\cap \{v_{ϵ}>0\}}{g}^{ij}{\partial }_i{v}_{ϵ}{\partial }_j{v}_{ϵ}={\int }_{(\partial B_R)\cap \Gamma }u{g}^{ij}(u-ϵF{)}_i{\nu }_j.$$

$$\frac{1}{\sqrt{1+c^2}}\left(J(u,B_R)-J(v_{ϵ},B_R)\right)=ϵ{\int }_{(\partial B_R)\cap \Gamma }u{g}^{ij}{F}_i{\nu }_j+\text{vol}(0<u<ϵF).$$

$${\int }_{(\partial B_R)\cap \Gamma }u{g}^{ij}{F}_i{\nu }_j$$

$$={\int }_{\partial (B_R\cap \Gamma )}(u{g}^{ij}{F}_i{\nu }_j-F{g}^{ij}{u}_i{\nu }_j)-{\int }_{(\partial \Gamma )\cap B_R}F$$

$$=-{\int }_{(\partial \Gamma )\cap B_R}F.$$

$$\frac{1}{\sqrt{1+c^2}}\left(J(u,B_R)-J(v_{ϵ},B_R)\right)=-ϵ{\int }_{(\partial \Gamma )\cap B_R}fd\sigma +\text{vol}(0<u<ϵF).$$

$$u_i+{\varphi }_i{u}_{x_3}=0;u_{ij}+u_{i{x}_3}{\varphi }_j+{\varphi }_{ij}{u}_{x_3}+{\varphi }_i{u}_{x_{3}j}+{\varphi }_i{\varphi }_j{u}_{x_3{x}_3}=0.$$

$$u_{ij}(P)={\varphi }_{ij}(P),\text{ for }i,j=x_1,x_2.$$

$$u_{x_1{x}_1}+u_{x_2{x}_2}=H.$$

$$({\partial }_j{g}^{ij})u_i+g^{ij}{u}_{ij}=0.$$

$$\frac{1}{1+c^2}{u}_{x_1{x}_1}(P)+u_{x_2{x}_2}(P)+u_{x_3{x}_3}(P)=0.$$

$$H=-u_{x_3{x}_3}(P).$$

Full text

1. Introduction

We study solutions to the problem

[TABLE]

on right circular cones in ${\mathbb{R}}^{n}$. We are interested in determining when the free boundary $\partial\{u>0\}$ is allowed to pass through the vertex of the cone.

The above problem has applications to two dimensional flow problems as well as heat flow problems (see [4] where (1.1) was first studied). When considering the applications on a manifold, one studies a variable coefficient problem in divergence form:

[TABLE]

Solutions of (1.2) may be found inside a bounded domain $\Omega$ by minimizing the functional:

[TABLE]

However, since the functional is not convex, minimizers of (1.3) may not be unique and there exist solutions to (1.2) which are not minimizers of (1.3). When the coefficients $a^{ij}(x)$ are Lipschitz continuous and satisfy an ellipticity condition, regularity of the free boundary was studied in [15]. The authors in [15] adapted the sup-convolution approach of Caffarelli in [7, 8, 9] for viscosity solutions. This approach relies on a nondivergence structure and therefore requires Lipschitz continuity of the coefficients $a^{ij}(x)$ so that (1.3) can be transformed into a nondivergence operator. More recently, the regularity of the free boundary for Hölder continuous coefficients $a^{ij}(x)$ was accomplished in [11] using different techniques. For coefficients $a^{ij}(x)$ assumed merely to be bounded, measurable, and satisfying the usual ellipticity conditions, regularity of the solution and its growth away from the free boundary was studied in [14]. However, to date nothing is know regarding the regularity of the free boundary when the coefficients $a^{ij}(x)$ are allowed to be discontinuous. In this paper we are interested in how the free boundary interacts with isolated discontinuous points of the coefficients $a^{ij}(x)$. In the context of a hypersurface, these points are considered to be a topological singularity. The simplest such case is the vertex of a cone. The aim of this paper is to study when the free boundary of a solution that arises as a minimizer is allowed to pass through a topological singularity. Before stating the main results of this paper we first recall a connection between solutions to (1.1) and minimal surfaces in order to understand what results one might expect for the free boundary problem on a cone.

1.1. Connection to minimal surfaces

Results for the singular set of free boundary points are analogous to results for the singular set of minimal surfaces. In the case of area-minimizing surfaces, the study of the singular set is reduced to considering area-minimizing cones. Simons [20] showed that any area-minimizing cone in ${\mathbb{R}}^{n}$ for $n\leq 7$ is necessarily planar. Simons actually proved a stronger result in [20] by showing that any minimal stable cone is planar. He also provided an example of a cone in ${\mathbb{R}}^{8}$ that is stable and therefore a possible candidate for being an area-minimizing cone. One year later, it was shown that the Simons cone is indeed area-minimizing, see [6]. As a consequence, $n=8$ is the first dimension for which a singularity of an area-minimizing hypersurface may occur.

Regarding the singular set of the free boundary for minimizers, the authors in [4] showed there are no singular points in dimension $n=2$. In [22] a monotonicity formula is utilized to show that blow-up solutions are homogeneous, and therefore the free boundary of blow-up solutions is a cone. As a further consequence there exists a minimal dimension $k^{*}$ such that the singular set of the free boundary of minimizers is empty if the dimension $n<k^{*}$. The authors in [10] showed $k^{*}>3$ and also provided an example of a nontrivial stable solution in dimension $n=7$. This example is analogous to the Simons cone and was later shown in [12] to indeed be a minimizer. Recently, the article [17] improved $k^{*}>4$. It is still an open problem as to whether $k^{*}=5,6,$ or $7$.

The article [21] further strengthened the connection between minimal surfaces and the free boundary problem by establishing a one-to-one correspondence between solutions of (1.1) in ${\mathbb{R}}^{2}$ and minimal bigraphs in ${\mathbb{R}}^{3}$. The one-to-one correspondence further strengthens the principle of a reduction in one dimension when moving from the theory of minimal surfaces to the one phase problem. Recall for instance that $k^{*}=8$ for area-minimizing surfaces where as $k^{*}$ is most likely $7$ for minimizers of (1.3).

1.2. Area-minimizing surfaces and the free boundary problem on cones

In light of the connection described above, one may expect that results for the free boundary problem on cones are analogous to the results for area-minimizing surfaces on cones. The description of the free boundary problem on cones (2.3) as well as the corresponding functional (2.2) is given later in Section 2.

On two-dimensional cones area-minimizing surfaces of co-dimension $1$ are distance minimizing geodesics. Two-dimensional cones in ${\mathbb{R}}^{3}$ are determined by the intersection of the cone with the two-sphere. If this intersection is a simple closed curve $\gamma$ on $S^{2}$, then the following Proposition is well known.

Proposition 1.1**.**

If length$(\gamma)<2\pi$, no distance minimizing geodesics pass through the vertex. If length $(\gamma)\geq 2\pi$, then there are distance minimizing geodesics that pass through the vertex.

The first statement in Proposition 1.1 can be found in Section 4-7 of [13]. The author with Chang Lara proved the following complete analogous result [2] for minimizers of (2.2) on a two-dimensional cone.

Theorem 1.2**.**

If $u$ is a minimizer of (2.2) on a two-dimensional cone, and if length$(\gamma)<2\pi$, then the vertex of the cone $0\notin\partial\{u>0\}$. If length$(\gamma)\geq 2\pi$, then the free boundary can pass through the vertex.

The proof of Theorem 1.2 was the main result in [2] and utilized that two-dimensional cones are isometrically flat. A competitor with a smaller functional value was constructed via an iterative argument that depended on length$(\gamma)$.

In this paper we consider the free boundary problem on higher dimensional cones. In view of the connection between area-minimizing surfaces and the one phase free boundary problem, one is led to ask about area-minimizing surfaces on higher dimensional cones. Morgan [18] considered area-minimizing surfaces on $n$-dimensional cones defined by

[TABLE]

with $c\geq 0$, and proved that a $k$-dimensional plane through the vertex is area-minimizing if and only if

[TABLE]

where $c=\delta^{-1}\sqrt{1-\delta^{2}}$. As a corollary (1.5) also determines when a $k$-dimensional area minimizing hypersurface can pass through a cone for $3\leq k\leq 7$. In particular, a $3$-dimensional area minimizing hypersurface may pass through the vertex of a $4$-dimensional cone as given in (1.4) and this is determined by (1.5). By the aforementioned drop in one dimension from area-minimizing surfaces to the free boundary problem, one may expect that the lowest dimension for which a free boundary of a minimizer may pass through the vertex of a cone of type (1.4) is for a three dimensional cone, and this depends on the constant $c$. We prove this is indeed the case.

1.3. Main Results

We prove results analogous to those of area-minimizing surfaces on cones in [18]. In Section 3 we establish a second variational formula for minimizers of (2.2) on a cone of type (1.4). With a notion of second variation one may discuss whether a solution is stable. Our first result regards the stability of a homogeneous solution.

Theorem 1.3**.**

Let $\mathcal{C}$ be a three-dimensional cone of type (1.4). There exists $0<c_{0}<\infty$ such that if $c\leq c_{0}$, then there exists a unique $($up to rotation$)$ $1$-homogeneous solution of (2.3) that is stable. If $c>c_{0}$ then no $1$-homogeneous solution of (2.3) is stable.

From the above theorem we obtain the following

Corollary 1.4**.**

Let $\mathcal{C}$ be a three-dimensional cone of type (1.4), and let $c_{0}$ be the constant in Theorem 1.3. If $c>c_{0}$ and $u$ is a minimizer of (2.2), then the vertex $0\notin\partial\{u>0\}$.

In the history of area-minimizing surfaces and free boundary problems it is common for stable solutions to indeed be global minimizers. Furthermore, the notion of stability and area-minimizing for hyperplanes on cones coincides [18]. Therefore, it is reasonable to assume that Theorem 1.3 may be improved by replacing the notion of stable with minimizer. In that vein we have our second main result.

Theorem 1.5**.**

Let $\mathcal{C}$ be a three-dimensional cone of type (1.4). There exists $c_{1}$ with $0<c_{1}\leq c_{0}<\infty$ such that if $c\leq c_{1}$, then there exists a minimizer $u$ of (2.2) such that $0\in\partial\{u>0\}$.

The significance of Theorem 1.3 is that $c_{1}>0$ which shows that three is the lowest dimension for which the free boundary of a minimizer passes through the vertex of a non-flat right circular cone. We expect that $c_{1}=c_{0}$ in Theorem 1.5, but a proof that $c_{1}=c_{0}$ may need to rely on numerical analysis such as in [12].

Many of the results in this paper apply to higher-dimensional cones of type (1.4). In Section 4 we present a symmetric $1$-homogeneous solution $\Phi_{c}$ to (2.3) on $\mathcal{C}$ where the vertex $0\in\partial\{u>0\}$. This symmetric solution $\Phi_{c}$ has a variant in each dimension. In Section 5 we show that if $v$ is any homogeneous stable solution with $0\in\partial\{v>0\}$, and $\mathcal{C}$ is three-dimensional, then $v\equiv\Phi_{c}$. Theorems 1.3 and 1.5 are reduced to showing whether the specific solution $\Phi_{c}$ is stable or is a minimizer. The symmetric candidate $\Phi_{c}$ is analogous to hyperplanes on cones as in [18]. As previously mentioned, there exist non-hyperplane cones that are area-minimizing; consequently, the results in [18] for hyperplanes can only be used to classify when an area-minimizing hypersurface of dimension $k$ passes through the vertex of a cone when $k\leq 7$. Similarly, in order to prove Theorems 1.3 and 1.5 on higher-dimensional cones, one would have to show any stable homogeneous solution $v\equiv\Phi_{c}$. Although we only present results in this paper for the symmetric solution $\Phi_{c}$ on a three-dimensional cone, the same techniques will apply in higher dimensions. We state this in the following remark which is a partial analogue to Theorem 1.1 in [18] regarding hyperplanes on cones.

Remark 1.6*.*

Let $\mathcal{C}$ be a cone of type (1.4). Let $\Phi_{c}$ be the symmetric solution as defined in Section 4. If $n\geq 3$, then there exists two constants $0<c_{1}\leq c_{0}<\infty$ depending on dimension $n$ such that $\Phi_{c}$ is stable if and only if $c\leq c_{0}$ and $\Phi_{c}$ is a minimizer of (2.2) if $c\leq c_{1}$.

Finally, one may consider Lipschitz manifolds with isolated singularities. Suppose $\mathcal{M}$ is a three-dimensional manifold and $\mathcal{M}$ is smooth in a neighborhood $\Omega\setminus\{x_{0}\}$. If there exists a sequence $r_{k}\to 0$ such that a rescaling $\mathcal{M}_{k}\to\mathcal{C}$ where $\mathcal{C}$ is of type (1.4), then the results of Theorem 1.5 will apply in a neighborhood of $x_{0}$. This is because if the free boundary of a minimizer $u$ passes through $x_{0}$, then using the regularity results in [14], one may obtain via a blow-up procedure a minimizer $u_{0}$ on $\mathcal{C}$ and the vertex $0\in\partial\{u_{0}>0\}$.

1.4. Outline and Notation

The outline of this paper is as follows. In Section 2 we define the notion of a solution to the free boundary problem and the corresponding functional. We also state some preliminary results necessary later in the paper. In Section 3 we give a second variation formula for $1$-homogeneous solutions. In Section 4 we present a symmetric $1$-homogeneous solution $\Phi_{c}$. In Section 5 we classify when $\Phi_{c}$ is a stable solution and prove Theorem 1.3 and Corollary 1.4. In Section 6 we show that for $c$ small but still positive the solution $\Phi_{c}$ is a minimizer of the functional (2.2) and thus prove Theorem 1.5.

We will use the following notation throughout the paper.

• •

$n$ refers to dimension.

• •

$\mathcal{C}$ is always a cone of type (1.4).

• •

$c$ is always the constant appearing in the definition of a cone in (1.4).

• •

$\nabla_{c}$ refers to the gradient on $\mathcal{C}$ arising from the inherited metric as explained in Section 2.

• •

$\Delta_{c}$ refers to the Laplace-Beltrami operator on $\mathcal{C}$ as explained in 2.

• •

$\nabla_{\theta}$ refers to the gradient on the sphere.

• •

$\Delta_{\theta}$ represents the Laplace-Beltrami on the sphere.

• •

$\Gamma:=\{u>0\}$ where $u$ is $1$-homogeneous solution to (2.3).

2. Preliminaries

We consider a $3$-dimensional cone $\mathcal{C}$ in ${\mathbb{R}}^{4}$ given by

[TABLE]

We study minimizers of the functional

[TABLE]

where $\nabla_{c}v$ is the gradient on the cone $\mathcal{C}$ away from the vertex. As shown later in Proposition 2.2, minimizers of (2.2) over $\Omega\subseteq\mathcal{C}$ are solutions to

[TABLE]

where $\nabla_{c}$ is the gradient on $\mathcal{C}$ from the inherited metric, and $\Delta_{c}$ is the Laplace-Beltrami on $\mathcal{C}$. The above class of solutions may seem restrictive; however, not only will all minimizers of (2.2) be solutions to (2.3), but also the common notion of viscosity solution as in [7, 8, 9] would also be a solution to (2.3) since the cone $\mathcal{C}$ is three-dimensional.

We consider two main parametrizations of the cone $\mathcal{C}$. Using spherical coordinates we have

[TABLE]

Under these coordinates the area form is $\sqrt{\text{det}(g)}=r^{2}\sin\phi\sqrt{1+c^{2}}$. The local coordinates $g^{ij}$ are given by

[TABLE]

and in these local coordinates we minimize

[TABLE]

Any minimizer will satisfy

[TABLE]

and so a minimizer of (2.2) is a solution to (2.3). We note that the first condition is written out as

[TABLE]

in the set $\{u>0\}$.

We may also work in the coordinates

[TABLE]

In these coordinates the area form is $\sqrt{g}=\sqrt{1+c^{2}}$ and the local coordinates $g^{ij}$ are given by

[TABLE]

We define $\displaystyle\mathcal{C}_{r}:=\{(y_{1},y_{2},y_{3},y_{4})\in\mathcal{C}:\sqrt{y_{1}^{2}+y_{2}^{2}+y_{3}^{2}}<r\}$. From the regularity results in [14], we have the following regarding the continuity of the minimizer as well as the growth away from a free boundary point at the vertex.

Proposition 2.1**.**

Let $u$ be a minimizer of (2.2) on $\Omega\subseteq\mathcal{C}$, then $u$ is Hölder continuous inside $\Omega$. Furthermore, if the vertex $\{0\}\in\partial\{u>0\}$, then there exists two constants $C_{1},C_{2}$ depending on $u$ such that

[TABLE]

We also have

Proposition 2.2**.**

Let $u$ be a minimizer of (2.2) in $\Omega$. Then $u$ is a solution to (2.3).

Proof.

From Proposition 2.1 minimizers are continuous. Then by considering variations in the positivity set, it follows that $\Delta_{c}u=0$ in $\{u>0\}$. Furthermore, the coefficients $g^{ij}$ are smooth away from the vertex of the cone. Therefore, when the cone is three-dimensional one may combine the results in [15] and [10] to conclude that $\partial\{u>0\}$ is smooth away from the vertex. It then follows from the domain variation techniques in [4] that the free boundary relation is satisfied away from the vertex so that any minimizer $u$ of (2.2) is a solution to (2.3). ∎

We also have the following Weiss-type monotonicity formula.

Proposition 2.3**.**

Let $u$ be a minimizer to (2.2) on $\Omega\subseteq\mathcal{C}$ and $\mathcal{C}_{R}\subset\Omega$. Assume that the vertex $0\in\partial\{u>0\}$. Then the functional

[TABLE]

is monotone increasing in $r$ for $r\leq R$. Furthermore, if $0<r_{1}<r_{2}\leq R$ and $W(r_{1},u)=W(r_{2},u)$ if and only if $u$ is homogeneous of degree $1$ on $\mathcal{C}_{r_{2}}\setminus\mathcal{C}_{r_{1}}$.

The proof of Proposition 2.3 relies on a radial domain variation. Since $\mathcal{C}$ may be parametrized by radial spherical coordinates, the usual proof will go through. For two-dimensional cones this was shown in [2]. The same proof has also been adapted for a more complicated weight (see [1, 3]). As a consequence of Propositions 2.1 and 2.3 we have the following

Proposition 2.4**.**

Let $u$ be a minimizer of (2.2) on $\Omega\subseteq\mathcal{C}$ and assume the vertex $0\in\partial\{u>0\}$. Then there exists a sequence $r_{k}\to 0$ and a $1$-homogeneous solution $u_{0}$ of (2.3) such that $0\in\partial\{u_{0}>0\}$ and the rescaled functions $u_{k}:=u(r_{k}x)/r_{k}$ converge uniformly on compact subsets of $\mathcal{C}$ to $u_{0}$. Furthermore, $u_{0}$ will be a minimizer of (2.2) on all $\Omega\Subset\mathcal{C}$.

Proof.

The following proof is standard. From Proposition 2.1, the Hölder estimates in [14], and the Arela-Ascoli Theorem, there exists $r_{k}\to 0$ and $u_{0}$ such that $u_{r_{k}}\to u_{0}$ locally uniformly. That $u_{0}$ is a minimizer of (2.2) and therefore also a solution to (2.3) is standard. From the rescaling property $W(\rho r,u)=W(\rho,u_{r})$ of the Weiss functional and the monotonicity of $W(r,u)$ it follows that $W(r,u_{0})=W(0+,u)$ for all $r$, and so $u_{0}$ is homogeneous of degree $1$. ∎

3. A Second Variation Formula

In this section we adapt the ideas in [10] to obtain a second variation formula. From Proposition 2.4 we restrict ourselves to the study of $1$-homogeneous solutions $u$ to (2.3). We denote the positivity set of a $1$-homogeneous solution $u$ to (2.3) by $\Gamma:=\{u>0\}$. The free boundary $\partial\Gamma$ is a cone and we denote the mean curvature by $H$. The main result in this section is the following Lemma that gives a second variation formula for $1$-homogeneous solutions to (2.3) that are also minimizers of (2.2). We define $\mathcal{F}_{R}:=\{F\in C^{\infty}(\overline{\Gamma}):F(x)=0\text{ if }|x|\notin(R^{-1},R)\}$ and $\mathcal{F}:=\cup\mathcal{F}_{R}$.

Lemma 3.1**.**

Let $\mathcal{C}$ be a cone of type (2.1). Let $u$ be a $1$-homogeneous minimizer of (2.2). Let $g^{ij}$ be the local coordinates from (2.6). Then

[TABLE]

Proof.

To prove Lemma 3.1 we intentionally follow the structure of the analogous Lemma in [10], so that the reader may compare.

We define $\Omega:=\Gamma\cap B_{R}$ for some fixed $R$, and use the local coordinates $g^{ij}$ as given in (2.6). We first assume that $F\in C_{0}^{\infty}({\mathbb{R}}^{3}\setminus\{0\})$ solves $\partial_{i}(\sqrt{g}g^{ij}F_{j})=0$ in $\Gamma$. Since $\Gamma$ is an NTA domain it is also a Twisted Hölder domain of order $1$ so by Theorem 4.5 in [5] there is a boundary Harnack inequality for $\Delta_{c}$ on $\Gamma$. Then there exists a constant $C$ such that $F\leq Cu$ in a neighborhood of the origin. We define

[TABLE]

Let $v_{\epsilon}=u-\epsilon F$ on $\overline{\Omega}_{\epsilon}$ and $v_{\epsilon}=0$ on $B_{R}\setminus\Omega_{\epsilon}$. We then have that $v_{\epsilon}=u$ on $\partial B_{R}$. Integrating by parts we have

[TABLE]

Since $\sqrt{g}=\sqrt{1+c^{2}}$ is a constant we may divide by $\sqrt{g}$ when minimizing the functional. We then have

[TABLE]

We note that in the above equation the volume element is from the flat metric since we have already divided out by $\sqrt{1+c^{2}}$. Integrating by parts on $\Omega$ we have

[TABLE]

Then we have that

[TABLE]

In [10] three remarks are given. The first remark is

Remark 3.2*.*

$u_{\nu\nu}=-H$ on $\partial\Gamma$ except at the origin.

Proof.

Under a rotation we assume our point $P\in\partial\Gamma$ to be $(x_{1},0,0)$. Locally near $P$ the free boundary is given by $u(x_{1},x_{2},\phi(x_{1},x_{2}))=0$. Differentiating with respect to $i,j$ (with $i,j=x_{1},x_{2}$) we obtain

[TABLE]

We now evaluate at $P$ and use that $\phi_{i}(P)=0$ and $u_{x_{3}}(P)=-1$ to conclude that

[TABLE]

Recalling that $H$ is the mean curvature of $\partial\Gamma$ at $P$ we have that

[TABLE]

Now from the local coordinates given in (2.6) for $i,j=x_{1},x_{2},x_{3}$ we have

[TABLE]

Evaluating at $P$ and using that $u_{x_{1}}(P)=u_{x_{2}}(P)$ as well as $x_{2}=x_{3}$ at $P$ we obtain

[TABLE]

We finally note that because $\partial\Gamma$ is a cone, that $(t,0,0)\in\partial\Gamma$ for all $t\geq 0$, so that $u_{x_{1}x_{1}}(P)=0$. Then combining the above equation with (3.3) we obtain that

[TABLE]

This concludes the proof of Remark 3.2. ∎

Remark 3.3*.*

$H\geq 0$. In particular, ${\mathbb{R}}^{3}\setminus\Gamma$ is a finite union of convex cones.

Whereas the proofs of Remarks 3.2 and 3.4 are very similar to those found in [10], the proof of Remark 3.3 is different and requires Lemma A.2 from the Appendix.

Proof of Remark 3.3.

We utilize the homogeneity of $u$. Since $u=rf$ and $\Delta_{c}u=0$ whenever $u>0$ it follows that $\Delta r^{\alpha}f=0$ as long as

[TABLE]

Notice that $0<\alpha<1$ for $c>0$. From Lemma A.2, we have $|\nabla_{\theta}f|^{2}$ achieves the maximum on $S^{2}\cap\partial\Gamma$. Since $|\nabla_{\theta}f|=1$ everywhere on $\partial\Gamma$, we conclude that $|\nabla_{\theta}f|\leq 1$ in $\Gamma$. Under a rotation we may assume that $(1,0,0)=P\in\partial\Gamma$ and $\phi$ is the outward unit normal from the spherical coordinates given in (2.4). Then $\partial_{\phi}f_{\phi}^{2}=2f_{\phi}f_{\phi\phi}\geq 0$, and since $f_{\phi}(P)=-1$ we obtain that $f_{\phi\phi}(P)\leq 0$. Then $u_{33}\leq 0$, and it follows that $H\geq 0$. ∎

Remark 3.4*.*

Let $z:U\to{\mathbb{R}}^{3}$ defined on an open subset $U$ or ${\mathbb{R}}^{2}$ be a local parametrization of the surface $\partial\Gamma$ and let $\nu=\nu(s)$ be the unit normal to $\partial\Gamma$ at $z(s)$ pointing away from $\Gamma$. In the coordinate system $x(s,t)=z(s)-t\nu(s)$, the volume element $dV=(1+tH+O(t^{2}))d\sigma(s)dt$, where $d\sigma(s)$ is the $n-1$ area element on the surface.

The volume element $dV$ in Remark 3.4 is for the flat metric; therefore, Remark 3.4 is identical to that in [10] and the same proof applies.

We now finish the proof of Lemma 3.1. Note that in the local coordinates (2.6), if $P\in\partial\Gamma$, then $FF_{\nu}=g^{ij}F_{i}\nu_{j}$ at $P$ where $\nu$ is the outward unit normal at $P$. This is most easily seen by under a rotation letting $P=(x_{1},0,0)$. Now by combining Remarks 3.2, 3.3, and 3.4 as in [10] we obtain

[TABLE]

Since $u$ is a minimizer of $J$ we have that

[TABLE]

so that

[TABLE]

Since the above inequality is true for all $\Delta_{c}F=0$ it follows that

[TABLE]

for any $F\in C_{0}^{\infty}({\mathbb{R}}^{3}\setminus\{0\})$. This concludes the proof of Lemma 3.1. ∎

We recall the following Lemma from [10].

Lemma 3.5**.**

Suppose that $\Gamma$ is an open cone in ${\mathbb{R}}^{3}$ and $\partial\gamma=(\partial\Gamma)\cap S^{2}$ is a finite untion of smooth curves. Then the mean curvature $H$ of $\partial\Gamma$ can be written as

[TABLE]

where $\kappa$ is the geodesic curvature of the curve $\gamma$ in the unit sphere $S^{2}$.

Using Lemma 3.5 we may prove

Lemma 3.6**.**

Let $\Gamma$ be a cone in ${\mathbb{R}}^{3}$ with the mean curvature of $H$ of $\partial\Gamma$ satisfying $H\geq 0$ as well as (3.1) for every $F\in C_{0}^{\infty}({\mathbb{R}}^{3}\setminus\{0\})$. Then $\Gamma^{c}$ has one connected component and is a convex cone contained in a half space.

Proof.

We choose $F$ to be a radial function. We let $\gamma:=\partial\Gamma\cap S^{2}$ and $U:=S^{2}\cap\Gamma$. Then

[TABLE]

where $\alpha(s)$ is a parametrization of $\gamma$ with respect to arc length. We now convert the integral over $\Gamma$ from the local coordinates given by (2.6) to the spherical coordinates given by (2.4).

[TABLE]

and $|U|$ is the area of $U$ in the unit sphere. Using the same choice of radial function $F(|r|)$ as in [10], and combining the above two inequalities we obtain

[TABLE]

We now label $V_{j},\ j=1,\ldots,m$ as the connected components of $S^{2}\setminus U$ and $\gamma_{j}$ as the boundary curves. Using that $H\geq 0$, so that $\kappa\geq 0$, we apply the Gauss-Bonnet formula

[TABLE]

and sum over $j$ to obtain

[TABLE]

Since $\displaystyle\sum_{j=1}^{m}|V_{j}|=4\pi-|U|$, we have that

[TABLE]

Then $m=1$, and $U^{c}$ is a single connected convex component of $S^{2}$, and so $U^{c}$ must be contained in a half space. ∎

In this Section we have closely followed the ideas in [10]. Moving forward, however, the ideas in this paper are very different. When $c=0$ one may use a homogeneity argument to conclude that $\partial\Gamma$ is flat. When $c>0$ we will see in the next Section that there exists a stable solution $\Phi_{c}$ where $\partial\Gamma$ is not flat. Moreover, in Section 6 we show that for $c>0$ and small enough that these candidate solutions are indeed minimizers.

4. The Symmetric Solution

In this Section we present a homogeneous solution $\Phi_{c}$ which will turn out to be stable and even a minimizer for certain values of $c$. We also show that up to rotation $\Phi_{c}$ is the only possible $1$-homogeneous stable solution. From Lemma 3.5 if $u$ is a $1$-homogeneous solution that is stable, then $\{u=0\}$ consists of a single connected component that is convex and contained in a half space. For each $c>0$ we now describe a symmetric candidate solution. Using the spherical coordinates (2.4), if $\Delta_{c}r^{\beta}f=0$ and $f(\theta,\phi)$ is a function of $\phi$ alone and independent of $\theta$, then

[TABLE]

Under the change of variables $t=\cos\phi$, the function $f$ is a Legendre function and well understood. For $c\geq 0$ and fixed $\beta=1$ there is a unique solution with $f^{\prime}(\phi)=0$ and $f^{\prime}(\phi_{0})=1$ where $f(\phi_{0})=0$ which we will denote by $f_{1,c}$. We note that $f_{1,c}(\phi)\geq 0$ for $\phi\leq\pi/2$. Then $rf_{1,c}(\phi)$ is a symmetric (in the variable $\theta$) candidate solution which we will denote by $\Phi_{c}(r,\phi)=rf_{1,c}(\phi)$. We now show that if $u$ is a stable $1$-homogeneous solution, then up to rotation $u\equiv\Phi_{c}$.

Lemma 4.1**.**

Let $u$ be $1$-homogeneous solution to (2.3) and assume that $\{u=0\}$ is a single connected component contained in a half space. Then up to rotation $u\equiv\Phi_{c}$.

Proof.

We will use the moving plane method as presented in [19]. By rotation we will assume that $\{u=0\}$ is contained in the hemisphere given by $0\leq\phi\leq\pi/2$. Let $u=rg(\theta,\phi)$. We claim that for $0\leq\phi\leq\pi/2$ we have $g(\theta,\phi)\leq g(\theta,\pi-\phi)$. That is when $g$ is reflected across the equator the bottom half is always greater than or equal to the upper half. Let $h(\theta,\phi)=g(\theta,\pi-\phi)$ and consider $\{g>h\}\cap\{0\leq\phi\leq\pi/2\}$. Notice that $(g-h)(\theta,\phi/2)=0$ and is a nonnegative eigenfunction $\Delta_{\theta}(g-h)=-\lambda(g-h)$ on $\{g>h\}\cap\{0\leq\phi\leq\pi/2\}$. Now $|\{g-h\leq 0\}\cap\{0\leq\phi\leq\pi/2\}|>0$, so that

[TABLE]

Both $g-h$ and $h$ are both positive eigenfunctions on their respective positivity sets with the same eigenvalue. If $\{g-h>0\}\cap\{\phi\leq\pi/2\}\neq\emptyset$, then the eigenvalue for $g-h$ would be strictly larger than the eigenvalue for $h$ which would be a contradiction. Therefore, $g\leq h$ whenever $\phi\leq\pi/2$.

With this comparison principle in place one can begin to rotate the equator into the set $\{g=0\}$. We have the same comparison argument as before, so that the reflection will always lie below. Then conditions $(A),(B),(C),(D)$ of Section 3 in [19] are all met, so that we conclude $\{g=0\}$ is a spherical cap. Now the nonnegative eigenfunction on a domain is unique, and after rotation there is a nonnegative eigenfunction given by (4.1). Since $u=rg$ and $\Delta_{c}u=0$ it follows that $\beta=1$ so that $g=f_{1,c}$, so that $u=\Phi_{c}$.

∎

5. Stable Solutions

In this Section we prove Theorem 1.3. We first show that for $c$ large enough the solution $\Phi_{c}$ is not stable.

Lemma 5.1**.**

There exists $M<\infty$ such that if $c\geq M$, then $\Phi_{c}$ is not a stable solution.

Proof.

Let $\phi_{0}$ be such that $f_{1,c}(\phi_{0})=0$. We also label $t_{0}=\cos\phi_{0}$. We have $t_{0}\to-1$ as $c\to\infty$. Furthermore, the mean curvature $H$ of $\partial\Gamma$ is $H_{1}/r$ where $r$ is the distance from the origin, and $H_{1}$ is the mean curvature at radius $1$ which is

[TABLE]

We choose a radial function $F(r)$ which is smooth and compactly supported in $B_{1}\setminus\{0\}$. Then

[TABLE]

Thus, the above quantity remains constant independent of $c$. Furthermore, by converting the local coordinates from (2.6) to the spherical coordinates (2.4) we have

[TABLE]

As $c\to\infty$, the above quantity goes to zero. Therefore, for large enough $c$ the second variational inequality (3.1) fails, and we conclude that our candidate solution $\Phi_{c}$ is not stable. ∎

For this next Lemma we define $G_{\beta}(r,\phi):=r^{\beta}g_{c}(\phi)$ where $g_{c}(\phi)$ is a solution to (4.1) with $-1<\beta<0$ and the conditions $g_{c}^{\prime}(0)=0$ and $g_{c}(\phi)\geq 0$ for all $0\leq\phi<\pi$. Now $G_{\beta}^{\prime}\geq 0$ as well. We will see in the proof that it is convenient to choose $\beta=-1/2$ in which case we will simply write $G$. Also, when the value of $c$ is fixed we will write simply $g(\phi)$. The function $g(\phi)$ should not be confused with the local coordinates $g^{ij}$ from the metric defined by (2.6).

Lemma 5.2**.**

Let $\Phi_{c}$ be the symmetric solution as described in Section 4. Let $\Gamma=\{\Phi>0\}$, and $\phi_{0}$ such that $\Phi_{c}(\phi_{0})=0$. Then (3.1) is equivalent to

[TABLE]

where $H_{1}$ is the mean curvature of $\Gamma$ at $r=1$, and $g(\phi)$ is a nonnegative solution to (4.1) with

[TABLE]

Proof.

We first show that (5.1) implies (3.1). Let $F\in\mathcal{F}$ with $F=0$ on $\partial B_{R_{2}}\cap\Gamma$ and $\partial B_{R_{1}}\cap\Gamma$. We aim to minimize the quantity

[TABLE]

From the standard theory of Calculus of variations, a unique minimizer $v$ will exist and satisfy the Steklov eigenvalue problem

[TABLE]

where $\lambda=E(v,\Gamma)$. In order to obtain a lower bound for $\lambda$, we consider the functions $G_{\beta}$. Since $G_{\beta}>0$ on $\partial\Gamma\cap(B_{R_{2}}\setminus B_{R_{1}})$, there exists some constant $M$ and a point $x_{0}\in\overline{\Gamma\cap(B_{R_{2}}\setminus B_{R_{1}})}$ such that $MG_{\beta}\geq v$ and $MG_{\beta}(x_{0})=v(x_{0})$. Since $\Delta_{c}G_{\beta}=0$ in $\Gamma$, then from the maximum principle we conclude $x_{0}\in\partial\Gamma\cap(B_{R_{2}}\setminus B_{R_{1}})$. Then if $x_{0}=(r_{0},\theta_{0},\phi_{0})$ and $H_{1}$ is the mean curvature of $\Gamma$ at $x_{0}/|r|$ we obtain

[TABLE]

Thus

[TABLE]

Then if

[TABLE]

then also $E(v,\Gamma)\geq 1$, and so (3.1) holds.

We now show that (3.1) implies (5.1). From the symmetry of $\beta$ for $-1<\beta<0$ in (4.1), one may expect a maximum bound from below when $\beta=-1/2$. We now show this is indeed the case. Recall that $G(r,\phi)=r^{-1/2}g(\phi)$ where $g(\phi)$ is a solution to (4.1) with $\beta=-1/2$. Then in local coordinates (2.6) we have

[TABLE]

since

[TABLE]

because $G(r,\phi)=r^{-1/2}g(\phi)$. This is also why we chose $\beta=-1/2$. Now

[TABLE]

We now define

[TABLE]

and note that

[TABLE]

Similarly if

[TABLE]

then

[TABLE]

Now if we define

[TABLE]

then

[TABLE]

Now as $R_{1}\to 0$ and $R_{2}\to\infty$ the first term in the last line above goes to zero. Hence, we conclude that

[TABLE]

Then (3.1) holds if and only if $H_{1}\leq g^{\prime}(\phi_{0})/g(\phi_{0})$. ∎

Corollary 5.3**.**

There exists $\epsilon_{0}>0$ such that if $c\leq\epsilon_{0}$, then $\Phi_{c}$ is a stable solution; i.e., if $\Gamma=\{\Phi_{c}>0\}$, then (3.1) holds.

Proof.

From Lemma (5.1) we have that (3.1) is holds if and only if

[TABLE]

As $c\to 0$ we have that $H_{1}\to 0$, $\pi/2\leq\phi_{0}\to\pi/2$, and $g_{c}\to\cos\phi$. Then the above inequality will be satisfied for small enough $c$. ∎

We are now ready to prove our first Main Theorem.

Proof of Theorem 1.3.

We first normalize by letting $g_{c}(0)=1$. If $0\leq c_{1}<c_{2}$, the $g_{c_{1}}$ is a supersolution to (4.1) for $c_{2}$, and so $g_{c_{1}}>g_{c_{2}}$ on $(0,\pi)$. Furthermore, for fixed $\phi_{0}$ if $g_{c_{1}}(\phi_{0})-h=g_{c_{2}}(\phi_{0})$, then $g_{c_{2}}-h$ is a subsolution on the interval $(\phi_{0},\pi)$ to (4.1) for $c_{2}$. Then $g_{c_{1}}-h>g_{c_{2}}$ on $(\phi_{0},\pi)$. It follows that $g_{c_{1}}^{\prime}>g_{c_{2}}^{\prime}$ on $(0,\pi)$. Then $g_{c_{1}}/g_{c_{2}}$ is increasing on $(0,\pi)$, so that $\log(g_{c_{2}}/g_{c_{1}})$ is increasing on $(0,\pi)$. Then

[TABLE]

Now if $g_{c_{2}}(\phi_{2})=0$, then the mean curvature of $\Gamma$ is

[TABLE]

Then (3.1) which is equivalent to (5.1) holds if and only if

[TABLE]

From Corollary 5.3 there exists $c_{2}>0$ such that $\Phi_{c_{2}}$ is a stable solution, so that (5.3) holds where $g_{c_{2}}(\phi_{2})=0$. We seek to show that if $0\leq c_{1}<c_{2}$, then (5.3) hols for $c_{1}$ where $g_{c_{1}}(\phi_{1})=0$. We take the derivative

[TABLE]

Since (5.3) holds at $\phi_{2}$, and since $\phi_{2}>\pi/2$, it follows that in a small neighborhood around $(\phi_{2}-\epsilon,\phi_{2}+\epsilon)$ that the above derivative is negative. Then if $c_{3}\in(c_{2}-\delta,c_{2}+\delta)$ for small enough delta, then $\phi_{3}\in(\phi_{2}-\epsilon,\phi_{2}+\epsilon)$ where $g_{c_{3}}(\phi_{3})=0$. Then if $c_{3}\in(c_{2}-\delta,c_{2})$ we have that

[TABLE]

The last inequality follows from (5.2). We have shown that the set of points $c\in[0,\infty)$ for which (5.3) holds is open to the left. Since the inequality is preserved in a limit, it follows that the set of points $c\in[0,\infty)$ for which (5.3) holds is also closed to the left. Combining this with Lemma 5.1 there is then a last point $c_{0}<\infty$ such that $\Phi_{c}$ is stable if and only if $0\leq c\leq c_{0}$. From Corollary 5.3 we have that $c_{0}>0$. This concludes the proof. ∎

We now give the

Proof of Corollary 1.4.

Let $u$ be a minimizer of (2.2) with $c>c_{0}$ with $c_{0}$ given in Theorem 1.3. Suppose by way of contradiction that the vertex $0\in\partial\{u>0\}$. By Proposition 2.4 there exists a $1$-homogeneous minimizer $u_{0}$ of (2.2). Then $u_{0}$ is also stable, and so by Lemmas 3.6 and 4.1 we conclude $u_{0}\equiv\Phi_{c}$. But $\Phi_{c}$ is not stable for $c>c_{0}$, and we obtain a contradiction. ∎

6. A minimizer for $c>0$.

In this section we prove that the symmetric solution $\Phi_{c}$ defined in Section 4 is indeed a minimizer for $0\leq c\leq c_{0}$ for $c_{0}$ small enough. This is accomplished by trapping $\Phi_{c}$ between a continuous family of sub- and supersolutions to the free boundary problem (2.3). This shows that $\Phi_{c}$ is a unique solution subject to its own boundary data. Since a minimizer does exist and is a solution, then $\Phi_{c}$ is a minimizer. We first construct a continuous family of subsolutions from below.

Lemma 6.1**.**

There exists $c_{0}>0$ such that if $0\leq c\leq c_{0}$, and if $u$ is a solution to (2.3) with $u=\Phi_{c}$ on $\partial B_{1}$, then $u\geq\Phi_{c}$ in $B_{1}$.

Proof.

We let $\Phi_{c}=rf_{1,c}(\phi)$. For convenience throughout this proof we will simply write $f(\phi)$ in place of $f_{1,c}(\phi)$. We consider $(v_{\epsilon})_{+}$ where $v_{\epsilon}:=rf(\phi)-\epsilon r^{\beta}g(\phi)$ and

[TABLE]

Notice that

[TABLE]

By choosing $-1<\beta<0$ and $M$ large enough depending on $\beta$ and $c_{0}$, then $\Delta_{c}r^{\beta}g(\phi)\leq 0$ for $0\leq c\leq c_{0}$. Thus, $\Delta_{c}(v_{\epsilon})_{+}\geq 0$ independent of $\epsilon$. For convenience throughout the remainder of the proof we will write simply $v$ in place of $v_{\epsilon}$.

We note that

[TABLE]

Furthermore, on $\{v=0\}$ we have $rf(\phi)=\epsilon r^{\beta}g(\phi)$, so that on $\{v=0\}$ we obtain

[TABLE]

In order to use a comparison principle, we need $|\nabla_{c}v|^{2}>1$ on $\partial\{v>0\}$. We let $\phi_{0}$ be such that $f(\phi_{0})=0$. Notice that $\phi_{0}>\pi/2$. Furthermore, since $\epsilon r^{\beta}g(\phi)\geq 0$, then $v(r,\phi)=0$ only when $\phi<\phi_{0}$. Finally, we note that

[TABLE]

We now take the derivative in $\phi$ of the last expression in (6.1).

[TABLE]

We recall that

[TABLE]

Substituting this into the computed derivative, reorganizing terms, and dividing by $2$, we obtain that the derivative (divided by $2$) is the sum of the following three pieces

[TABLE]

We choose $\beta=-1/2$. We also have that $f^{\prime\prime}(\phi)<-\delta_{1}<0$ for $0\leq\phi\leq\phi_{0}$ and some $\delta_{1}>0$ depending on $c_{0}$ and independent of $c$ if $0\leq c\leq c_{0}$. Then choosing $M$ large enough depending on $c_{0}$, there exists a constant $\delta_{2}$ depending on $c_{0}$ such that for $0\leq c\leq c_{0}$ we have

[TABLE]

Then

[TABLE]

Choosing again $M$ large enough, there exists $C_{1}$ such that if $0\leq c\leq c_{0}$, then

[TABLE]

The parameter $M$ is now fixed. For $c_{0}$ small, the angle $\phi_{0}$ is close to $\pi/2$. Therefore, to control $III$ we choose $c_{0}$ small enough so that

[TABLE]

Then $III\leq 0$, and so $I+II+III\leq 0$. Furthermore, in the above proof it is clear that $I+II+III<0$ when $0<\phi<\phi_{0}$, so that $|\nabla_{c}v|^{2}>1$ on $\partial\{v=0\}$. Thus we have shown that $\Delta(v_{\epsilon})_{+}\geq 0$ and $|\nabla(v_{\epsilon})_{+}|>1$ on $\partial\{(v_{\epsilon})_{+}>0\}$, and this is independent of $\epsilon>0$.

Now let $u$ be a solution to (2.3) with $u=\Phi_{c}$ on $\partial B_{1}$. Suppose by way of contradiction that there exists $x_{0}\in B_{1}$ such that $u(x_{0})<\Phi_{c}(x_{0})$. We have that $v_{\epsilon}<u$ on $\partial B_{1}$ for all $\epsilon>0$, and we may choose $\epsilon$ large enough so that $v_{\epsilon}<u$ in $B_{1}$. Then $(v_{\epsilon})_{+}<u$ in $\{u>0\}$. Also, $\{(v_{\epsilon})_{+}>0\}\subset\{u>0\}$. We now continuously shrink $\epsilon$ until either $(v_{\epsilon})_{+}$ touches $u$ from below in $\{u>0\}$, or $\partial\{(v_{\epsilon})_{+}>0\}$ touches $\partial\{u>0\}$. Since $(v_{\epsilon})_{+}\to\Phi_{c}$ pointwise on $B_{1}\setminus\{0\}$, there exists an $\epsilon_{0}$ and $x_{1}\neq 0$ such that $(v_{\epsilon})_{+}\leq u$ in $B_{1}$ and either $(v_{\epsilon_{0}})_{+}(x_{1})=u(x_{1})>0$ or $x_{1}\in(\partial\{(v_{\epsilon_{0}})_{+}>0\}\cap\partial\{u>0\})$. Since $\Delta_{c}(v_{\epsilon_{0}})>0$ in $\{(v_{\epsilon_{0}})_{+}>0\}$, the first possibility is a violation of the comparison principle. Since $(v_{\epsilon_{0}})_{+}\leq u$, if $(v_{\epsilon_{0}})_{+}(x_{1})=u(x_{1})=0$, then since $|\nabla_{c}u(x_{1})|=1<|\nabla_{c}(v_{\epsilon_{0}})_{+}|$ we also obtain a contradiction. Therefore, if $0\leq c\leq c_{0}$ and if $u$ is a solution to (2.3) with $u=\Phi_{c}$ on $\partial B_{1}$, then $u\geq\Phi_{c}$. ∎

Lemma 6.2**.**

There exists $c_{0}>0$ such that if $0\leq c\leq c_{0}$, and if $u$ is a solution to (2.3) with $u=\Phi_{c}$ on $\partial B_{1}$, then $u\leq\Phi_{c}$.

Proof.

The beginning of the proof is similar to the proof of Lemma 6.1. We consider $v_{c}=rf_{1,c}(\phi)+\epsilon r^{\beta}g(\phi)$ with $\beta=-1/2$ and $g(\phi)=M-\cos\phi$. We will write $f$ in place of $f_{1,c}$ when $c$ is understood. We use the subscript $c$ on the function $v_{c}$ because later in the proof we will let $c$ vary.

On the set $\{v_{c}=0\}$ we have $rf(\phi)=-r^{\beta}g(\phi)$, so that once again we obtain that on $\{v_{c}=0\}$ we have

[TABLE]

One main difference from the proof of Lemma 6.1 is that now if $v_{c}(r,\phi)=0$, then $\phi\geq\phi_{0}$ where $f(\phi_{0})=0$. Using the same computations as in the proof of Lemma 6.1, we obtain by taking the derivative in $\phi$ that $|\nabla_{c}v_{c}(r,\phi)|^{2}<1$ provided that $\phi<\phi_{1}$ where $\phi_{1}>\phi_{0}$ and $\phi_{1}$ is determined by letting $c$ be small so that the third term $III$ in (6.3) is negative. We now fix $\phi_{2}$ with $\phi_{0}<\phi_{2}<\phi_{1}$. Notice that $\phi_{0}$ depends on $c$, but for small enough $c_{0}$, $\phi_{1}$ and $\phi_{2}$ will not depend on $c$ for $0\leq c\leq c_{0}$. For fixed $\epsilon_{0}$, let $r_{0}$ be such that

[TABLE]

Then

[TABLE]

We have that $|\nabla_{c}v_{c}|>1$ on $\partial\{v>0\}$ as long as $\phi\leq\phi_{2}$. We now redefine the function $v_{c}$ on $B_{r_{0}}$. We first notice that from (6.4), we may rescale by

[TABLE]

The rescaled function $v_{r_{0}}$ is defined on $B_{1/r_{0}}$. Therefore, we may assume without loss of generality, that $\epsilon=-f(\phi_{2})/g(\phi_{2})$ and that we are redefining the values on $B_{1}$. We now define $U:=B_{1}\cap\{x_{3}>\cos(\phi_{2})\}$ and also define

[TABLE]

Finally, we paste the two functions $\tilde{v}_{c}$ and $v_{c}$ by defining

[TABLE]

Using a compactness argument, we will show for $\phi_{2}$ fixed and small enough $c$, that $\Delta_{c}w_{c}\leq 0$ in $\{w_{c}>0\}$ and $|\nabla_{c}w_{c}|<1$ on $\partial\{w_{c}>0\}\cap B_{1}^{c}$ and $\partial\{w_{c}>0\}\cap B_{1}$. Now $\{w_{c}>0\}$ is a wedge-type domain at $\partial\{w_{c}>0\}\cap\partial B_{1}$, but because of the angle of the wedge we will see that $\partial\{w_{c}>0\}\cap\partial B_{1}$ can never touch the free boundary $\partial\{u>0\}$ of a solution $u$ to (2.3).

We first show that if $c=0$, we obtain the needed properties for $w_{0}$. Then using a compactness argument, we show that if $c$ is small enough, that $w_{c}$ will also have the needed properties. If $c=0$, then on $\partial B_{1}$ and $\phi<\phi_{2}$ we have that

[TABLE]

so that

[TABLE]

This is just a linear function, and we notice that

[TABLE]

Furthermore, on $\partial B_{1}$ with $x_{3}>\cos\phi_{2}$, if $\nu$ is the outward unit normal to $B_{1}$ we have

[TABLE]

as long as $M>2$. Then $\Delta w_{0}>0$ weakly in $\{w_{0}>0\}$.

Now $U$ is not a $C^{1}$ domain. However by Lemma B.3, if we let $c\to 0$, then $\tilde{v}_{c}\to\tilde{v}_{0}$ in $C^{1}$ on $\overline{U}$. Thus, for $c$ small enough we obtain that

[TABLE]

and

[TABLE]

Thus, for $c_{0}$ small enough and for $0\leq c\leq c_{0}$, we have $\Delta_{c}w_{c}<0$ weakly in $\{w_{c}>0\}$, and a comparison principle holds.

We now let $c_{0}$ be chosen as above. Let $0\leq c\leq c_{0}$ and let $u$ be a solution to (2.3) with $u=\Phi_{c}$ on $\partial B_{\rho}$ with $\rho<-\cos\phi_{2}$ where $\phi_{2}$ defines $U$. Suppose that there exists $x_{0}\in B_{\rho}$ such that $u(x)>\Phi_{c}(x)$. We may choose $\epsilon$ large enough so that $w_{\epsilon}>u$ on $\overline{B}_{\rho}$. Since $w_{\epsilon}>\Phi_{c}$ in $B_{1}$ for every $\epsilon>0$, then also $w_{\epsilon}>u$ on $\partial B_{\rho}$ for every $\epsilon>0$. By continuously moving $\epsilon$ towards [math], there exists $\epsilon_{1}$ and $x_{1}\neq 0$ such that $w_{\epsilon_{1}}\geq u$ and either $w_{\epsilon_{1}}(x_{1})=u(x_{1})>0$ or $x_{1}\in\partial\{w_{\epsilon_{1}}>0\}\cap\{u>0\}$. If $w_{\epsilon_{1}}(x_{1})=u(x_{1})>0$, we obtain a contradiction since $\Delta_{c}w_{\epsilon_{1}}<0$ weakly in $\{w_{\epsilon_{1}}>0\}$. If $x_{1}\in\partial\{w_{\epsilon_{1}}>0\}\cap\{u>0\}$ and $|x_{1}|\neq 1$, we again obtain a contradiction since $w_{\epsilon_{1}}\geq u$, $|\nabla w_{\epsilon_{1}}(x_{1})|<1$, and $|\nabla u(x_{1})|=1$. We now consider the last case in which $|x_{1}|=1$ and $x_{1}\in\partial\{w_{\epsilon_{1}}>0\}\cap\{u>0\}$. Since $|\nabla v_{\epsilon_{1}}|<1$ on $\partial B_{1}\cap\partial\{v_{\epsilon_{1}}>0\}$ and $|\nabla\tilde{v}_{\epsilon_{1}}|<1$ on $\partial B_{1}\cap\{\tilde{v}_{\epsilon_{1}}>0\}$. Then

[TABLE]

for $x_{1}\in\partial B_{1}\cap\partial\{w_{\epsilon_{1}}>0\}$ and for some constant $0<\delta_{3}<1$ and $t$ small enough. We then again obtain a contradiction since $w_{\epsilon_{1}}\geq u$. Therefore, $u\leq\Phi_{c}$ on $B_{\rho}$. Since $\Phi_{c}$ is homogeneous, then by rescaling it is also true that $u\leq\Phi_{c}$ for any solution $u$ to (2.3) with $u=\Phi_{c}$ on $\partial B_{1}$. ∎

We now give the proof of our second main Theorem.

Proof of Theorem 1.5.

Let $0\leq c\leq c_{0}$. From the Calculus of Variations there exists a minimizer $u$ of (2.2) with $u=\Phi_{c}$ on $\partial B_{1}$. By Proposition 2.2 the minimizer $u$ is a solution to (2.3) in $B_{1}$. By Lemma 6.2 we have $u\leq\Phi_{c}$. The $c_{0}$ in the statement of Lemma 6.1 is greater than or equal to the $c_{0}$ in the statement of Lemma 6.2, so that from Lemma 6.1 we also have $u\geq\Phi_{c}$ in $B_{1}$. Then $u\equiv\Phi_{c}$, and therefore $\Phi_{c}$ is a minimizer. ∎

Appendix A A maximum principle

In order to prove nonnegative mean curvature of the free boundary of a homogeneous solution, we will need two Lemmas. If $v=rf(\theta)$ so that $v$ is homogeneous of degree $1$, then

[TABLE]

Consequently,

[TABLE]

If $u$ is homogeneous of degree [math], we have a similar result for the Hessian and gradient. Although the following Lemma is not difficult to show in all dimensions via induction, we only state and prove it for three dimensions.

Lemma A.1**.**

Let $u:\mathbb{R}^{3}\to\mathbb{R}$ and assume $u$ is homogeneous of degree [math], then

[TABLE]

Proof.

If $\lambda_{i}$ are the eigenvalues of the Hessian,

[TABLE]

which is invariant under rotation. Therefore, we may assume without loss of generality that $x_{0}\in S^{n-1}$ is $x_{0}=(1,0,0)$, so that under spherical coordinates $\theta=0$ and $\phi=\pi/2$. One may then explicitly compute that at $x_{0}$ we have

[TABLE]

Then at $x_{0}$ we obtain

[TABLE]

Since at $x_{0}=(1,0,0)$ we have

[TABLE]

we conclude that

[TABLE]

∎

We also have the following

Lemma A.2**.**

Let $f:S^{2}\to\mathbb{R}$ with $u=r^{\alpha}f$ such that $\Delta u=0$ in $\{u>0\}$. If $0<\alpha\leq 1$, then $|\nabla_{\theta}f|^{2}$ achieves its maximum on $S^{n-1}\cap\partial\{u>0\}$.

Proof.

Since $\Delta u=0$, then $\Delta_{\theta}f=-\alpha(\alpha+1)f$. If $v=f$, then

[TABLE]

The last inequality is a result of Lemma A.1. Then $|\nabla v|^{2}$ is subharmonic in $\{u>0\}$, and consequently achieves the maximum on the boundary. Since $|\nabla v|^{2}=|\nabla_{\theta}f|^{2}$, the conclusion of the Lemma is immediate. ∎

Appendix B $C^{1}$ convergence on a wedge-type domain

In this appendix we show $C^{1}$ convergence of $w_{\epsilon}\to w_{0}$ on $\overline{U}$ where $w_{\epsilon},w_{0},U$ are defined in the proof of Lemma 6.2. We recall that $U:=\{x\in B_{1}:x_{3}>\cos\phi_{2}\}$ where $\phi_{2}>\pi/2$ and was fixed in the proof of Lemma 6.2. It is clear that $w_{\epsilon}\to w_{0}$ in $C^{1}$ except at the corner $\{x\in\partial B_{1}:x_{3}=\cos\phi_{2}\}$. We handle the issue of the corner with a series of Lemmas.

Lemma B.1**.**

Let $V:=\{(x,y,z)\in{\mathbb{R}}^{3}:0\leq\arctan(y/x)\leq\theta_{0}<\pi\}$. If $\Delta u=0$ and $u=0$ on $\partial V$, and $|u|\leq C|x|$ for $|x|\geq 1$ and some $C>0$, then $u\equiv 0$.

Proof.

Let $v(r,\theta,x_{3})=r^{\lambda}\sin(\lambda\theta)$ where $\lambda\theta_{0}=\pi$. Then $\Delta v=0$ in $V$, $v=0$ on $\partial V$, and $v\geq 0$ in $V$. Let $w$ be the harmonic lifting of $u_{+}$ on $B_{R}\cap V$. We will choose $R$ large and use the boundary Harnack principle [16].

[TABLE]

Since $\lambda>1$, as $R\to\infty$ we obtain that $u_{+}\equiv 0$. The same argument applies to $u_{-}$. ∎

Lemma B.2**.**

Let $\Delta_{c}u=0$ in $U$ with $u=0$ on $\{x_{3}=\cos\phi_{2}\}\cap\overline{B}_{1}$ and $u\in C^{1}(\overline{U})$. Then there exists $C<\infty$ depending on $\displaystyle\|u\|_{C^{1}(\overline{U})}$ and $c_{0}$ but independent of $c$ if $0\leq c\leq c_{0}$ such that if $x_{0}\in\partial B_{1}\cap\{x_{3}=\cos\phi_{2}\}$, then

[TABLE]

Proof.

We first translate so that $x_{0}=0$ the origin. We will use compactness combined with a blow-up similar to the argument in the proof of Theorem 6.1 in [3]. Suppose by way of contradiction that no such $C$ exists. Then there exists $u_{j},c_{j},r_{j}$ with $\Delta_{c_{j}}u_{j}=0$ and $r_{j}\to 0$ such that if

[TABLE]

then

[TABLE]

We let

[TABLE]

We have the following

[TABLE]

Then $c_{j}\to c$, and $u_{r_{j}}\to u_{0}$ with

[TABLE]

where after rotation

[TABLE]

Since $a^{ij}$ is a constant coefficient matrix, a linear change of variables in only the $x_{2}$ and $x_{3}$ variables will give a new solution $\tilde{u}_{0}$ with $\Delta\tilde{u}_{0}=0$. For $0\leq c\leq c_{0}$ with $c_{0}$ small, the transformed domain will still be a wedge-domain $V$ with angle less than $\pi$. Then by Lemma B.1, it follows that $\tilde{u}_{0}\equiv 0$ so that $u_{0}\equiv 0$. This contradicts the fact that $\displaystyle\sup_{U\cap B_{1}}u_{0}=1$.

∎

Lemma B.3**.**

Let $u_{k}$ be a sequence of solutions to $\Delta_{c_{k}}u_{k}=0$ in $U$ with $u_{k}=0$ on $\{x_{3}=\cos\phi_{2}\}\cap\overline{B}_{1}$. Assume $\lim_{k\to\infty}c_{k}=0$ and r that $u_{k}\to v$ uniformly in $\overline{U}$ and $u_{k}\to v$ in $C^{1}(\partial B_{1}\cap\{x_{3}\geq\phi_{2}\})$. Then $u_{k}\to v$ in $C^{1}(\overline{U})$.

Proof.

Suppose by way of contradiction that there exist points $x_{k}\in\partial U$ such that $x_{k}\to x_{0}$ and

[TABLE]

for some fixed $\epsilon$. Let $|x_{k}-x_{0}|=r_{k}$. We rescale by

[TABLE]

and

[TABLE]

By Lemma B.2 we have that

[TABLE]

Now $\tilde{u}_{k}-v_{k}\to v_{0}$ uniformly with $\Delta v_{0}=0$. Furthermore, since $u_{k}\to v$ in $C^{1}(\partial B_{1}\cap\{x_{3}\geq\phi_{2}\})$ it follows that $v_{0}\equiv 0$ on $\partial V$ where $V$ is the domain obtained in the blowup and satisfies the assumptions of Lemma B.2. Then since $v_{0}$ has linear growth it follows from Lemma B.1 that $v\equiv 0$. Now from the $C^{1}$ convergence of $u_{k}-v_{k}$ away from the wedge of $V$ and (B.1) it follows that there exists $x_{0}\in\partial V\cap\partial B_{1}$ such that $|\nabla v_{0}|>\epsilon$. But this contradicts the fact that $v\equiv 0$. ∎