A Heintze–Karcher-Type Inequality for Hypersurfaces with Capillary Boundary

Abstract

In this paper, we establish a Heintze–Karcher-type inequality for hypersurfaces with capillary boundary of contact angle $\theta \in \big (0,\frac{\pi }{2}\big ]$ in the half-space or a half ball, by using solution to a mixed boundary value problem in Reilly type formula. Consequently, we give a new proof of Alexandrov-type theorem for embedded capillary constant mean curvature hypersurfaces with contact angle $\theta \in \big (0,\frac{\pi }{2}\big ]$ in the half-space or a half ball.

Appendix A: A Fredholm Alternative for the Mixed Boundary Value Problem

The purpose of the appendix is to present a detailed statement and proof of the Fredholm alternative for mixed boundary elliptic equation, which was brought up in [11] without proof.

For completeness, we present some existence and regularity results for mixed boundary value problems

$$\begin{aligned} {\left\{ \begin{array}{ll} {{\bar{\Delta }}} f=h \quad &{}\text {in }\Omega ,\\ f=0 \quad &{}\text {on }\Sigma ,\\ {{\bar{\nabla }}}_{{{\bar{N}}}} f+ f=g &{}\text {on }\textrm{int}(T), \end{array}\right. } \end{aligned}$$

(48)

which was proved by Lieberman [11, 12].

To facilitate the presentation, We recall the definition of weighted Hölder spaces. Set $d_\Gamma (x)=\textrm{dist}(x,\Gamma )$, $\Omega _\delta =\{x\in \Omega :d_\Gamma (x)>\delta \}$. For $a\ge 0$, $b\ge -a$, we define

$$\begin{aligned} {|f|}_a^{(b)}=\sup _{\delta >0}\delta ^{a+b}{|f|}_{a;\Omega _\delta }, \end{aligned}$$

where $|f|_{a;\Omega _\delta }$ is the standard norm on $\Omega _{\delta }$. We denote by $H_a^{(b)}$ the set of all functions f on $\Omega$ with finite norm $|f|_a^{(b)}$.

Theorem A.1

There exists a solution $f\in C^{2}( \Omega \cup \textrm{int} (T))\cap C^0({{\overline{\Omega }}})$ to (48) for all $h\in C^{\alpha }(\Omega \cup \textrm{int} (T))$, $g\in C^{1,\alpha }(\Omega \cup \textrm{int} (T))$.

Proof

See [11, Theorem 1]. $\square$

Theorem A.2

Assume $\theta \in (0,\frac{\pi }{2})$. Let $f\in C^{2}( \Omega \cup \textrm{int} (T))\cap C^{0}({{\overline{\Omega }}})$ be a solution to (48). Then for any $\lambda \in \left( 1,\frac{\pi }{2\theta }\right)$ and any noninteger $a>2$, we have

$$\begin{aligned} |f|_a^{(-\lambda )}\le C\bigg (|h|_{a-2}^{(2-\lambda )}+|g|_{a-1}^{(1-\lambda )}+|f|_0\bigg ). \end{aligned}$$

(49)

Proof

See [12, Theorem 4]. Here we only explain the admissible range $\left( 1,\frac{\pi }{2\theta }\right)$ for $\lambda$, which is not explicitly expressed in [12, Theorem 4].

Locally near every $x_0\in \Gamma$, up to a transformation, there exists a Cartesian coordinate $(x^1,\ldots ,x^{n+1})$, centered at $x_0$, such that the corresponding cylindrical coordinates $(r,\eta ,x')$ is given by

$$\begin{aligned} x_1=r\cos \eta ,\quad x_2=r\sin \eta ,\quad x'=\big (x^3,\ldots ,x^{n+1}\big ), \end{aligned}$$

with $0\le \eta \le \theta$. This follows from the definition of the wedge condition by Lieberman in [11], which is clearly satisfied by domains that we consider.

The key of the proof in [12, Theorem 4] is to find a Miller-type barrier function $\psi =r^\lambda \varphi (\eta )$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} {{\bar{\Delta }}} \psi =r^{\lambda -2}(\lambda ^2\varphi +\varphi _{\eta \eta })\le -c_1r^{\lambda -2}\quad &{}\text {for }\eta \in (0,\theta ),\\ {{\bar{\nabla }}}_{{{\bar{N}}}} \psi =-r^{\lambda -1}\varphi _{\eta }\ge c_1 r^{\lambda -1} &{}\text {for } \eta =0,\\ c_1\le \varphi \le 1 &{}\text {for } \eta \in [0,\theta ], \end{array}\right. } \end{aligned}$$

(50)

with a positive constant $c_1$. The Miller-type barrier $\varphi (\eta )$ is constructed from a perturbation of ${\tilde{\varphi }}(\eta )=\cos (\lambda \eta )$, which has a positive lower bound in $[0, \theta ]$ and satisfies (50) only if $\theta \in (0,\frac{\pi }{2\lambda })$. Hence to ensure $\lambda >1$, one needs to restrict $\theta \in (0,\frac{\pi }{2})$. $\square$

Lemma A.1

If $f\in C^{2}( \Omega \cup \textrm{int} (T))\cap C^{0}({{\overline{\Omega }}})$ is a solution to (48). Then there exists a positive constant C such that

$$\begin{aligned} |f|_0\le C(|h|_0+|g|_0). \end{aligned}$$

(51)

Proof

Set $C_1=|f|_0+|g|_0$. For the case $B=\overline{{\mathbb {R}}^{n+1}_+}$, assume without loss of generality that $\Omega \subset \left\{ 0<x_{n+1}<R\right\}$ for some R large, consider the functions $v_1(x)=C_1(e^{x_{n+1}}-e^R)$. A direct computation then yields,

$$\begin{aligned} {\left\{ \begin{array}{ll} {{\bar{\Delta }}} v_1\ge C_1 \quad &{}\text {in }\Omega ,\\ v_1\le 0 \quad &{}\text {on }\Sigma ,\\ {{\bar{\nabla }}}_{{{\bar{N}}}} v_1+v_1\le -C_1 &{}\text {on } \textrm{int}(T). \end{array}\right. } \end{aligned}$$

(52)

Applying the maximum principle (see for example [22, Lemma 4.1]) to $f+v_1$ and $-f+v_1$, we obtain (51).

For the case $B=\overline{{\mathbb {B}}^{n+1}}$, we choose $v_2=C_1(|x|^2-4)$ to replace $v_1$, and the process follows similarly. $\square$

We can now derive a Fredholm alternative for the mixed boundary value problem.

Theorem A.3

Assume $\theta \in \big (0,\frac{\pi }{2}\big )$, let a be a noninteger greater than 2, $\lambda$ be defined as in Theorem A.2. Then either (a) the homogeneous problem

$$\begin{aligned} {\left\{ \begin{array}{ll} {{\bar{\Delta }}} f=0 \quad &{}\text {in }\Omega ,\\ f=0 \quad &{}\text {on }\Sigma ,\\ {{\bar{\nabla }}}_{{{\bar{N}}}} f-\gamma f=0 &{}\text {on }\textrm{int}(T), \end{array}\right. } \end{aligned}$$

(53)

has nontrivial solutions; or (b) the homogeneous problem has only the trivial solution, in which case, for all ${h}\in H_{a-2}^{(2-\lambda )}$, $g\in H_{a-1}^{1-\lambda }$ the inhomogeneous problem

$$\begin{aligned} {\left\{ \begin{array}{ll} {{\bar{\Delta }}} f=h \quad &{}\text {in }\Omega ,\\ f=0 \quad &{}\text {on }\Sigma ,\\ {{\bar{\nabla }}}_{{{\bar{N}}}} f-\gamma f=g &{}\text {on }\textrm{int}(T), \end{array}\right. } \end{aligned}$$

(54)

has a unique solution $f\in H_a^{(-\lambda )}$.

Remark A.1

By the monotonicity of norm we have $|f|_\lambda ^{(-\lambda )}\le |f|_a^{(-\lambda )}$. Therefore if there is a solution $f\in H_{a}^{(-\lambda )}$ to problem (54), then the solution f is also in $C^{1,\alpha }({{\overline{\Omega }}})$ with $\alpha =\lambda -1$.

Proof of Theorem A.3

We follow the classical proof of the Fredholm alternative for the Dirichlet problem but with a careful choice of function space. Set ${\mathcal {A}} =\{u\in H_a^{(-\lambda )}: f=0\text { on }\Sigma \}$, ${\mathcal {B}}= H_{a-2}^{(2-\lambda )}\times H_{a-1}^{(1-\lambda )}$. By Theorem A.1, Theorem A.2 and (51), there exists a solution $u\in {\mathcal {A}}$ to problem (48) for all $(h,g)\in {\mathcal {B}}$. Uniqueness of the solution to problem (48) follows from the maximum principle immediately. Let $Q:{\mathcal {B}}\rightarrow {\mathcal {A}}$ be an operator such that Q(h, g) is the unique solution of problem (48). One can readily see that the operator Q is well-defined and bijective. Notice that the inhomogeneous problem (54) is equivalent to the following equation

$$\begin{aligned} f-Q(0,(1+\gamma )f)=Q(h,g). \end{aligned}$$

(55)

Thus $f\in {\mathcal {A}}$ is a solution to (55) if and only if it solves (54). Let P be an operator from $H_{a-1}^{(1-\lambda )}$ to itself such that $Pu=Q(0,(1+\gamma )u)$ for all $u\in H_{a-1}^{(1-\lambda )}$. Then the equation (55) reads as

$$\begin{aligned} f-Pf=v, \end{aligned}$$

(56)

where $v=Q(h,g)$.

To apply the classical Fredholm alternative (see e.g., [5, Theorem 5.11]) for (56), we need to verify that P is a compact operator. To this end, let $\{g_k\}$ be a bounded sequence in $H_{a-1}^{(1-\lambda )}$. By Theorem A.1 and (51), there exists $f_k\in {\mathcal {A}}$ such that $Pg_k=f_k$, and we have

$$\begin{aligned} |f_k|_a^{(-\lambda )}\le C\left( |g_k|_{a-1}^{(1-\lambda )}+|f_k|_0\right) . \end{aligned}$$

(57)

Since $\left\{ |f_k|_0\right\} _k$ is bounded, we have $\left\{ f_k\right\} _k$ is also bounded in $H_{a}^{(-\lambda )}$. By virtue of the Ascoli-Arzela theorem, up to a subsequence, there exists ${{\tilde{f}}}\in H_{a-1}^{(1-\lambda )}$, such that $f_{k_j}$ converges to ${{\tilde{f}}}$ in $H_{a-1}^{(1-\lambda )}$. Thus P is compact and the classical Fredholm alternative applies: (56) has a unique solution $f\in H_{a-1}^{(1-\lambda )}$, provided that the homogenous equation $f-Pf=0$ has only the trivial solution $f=0$. Since Q maps ${\mathcal {B}}$ onto ${\mathcal {A}}$, any solution $f\in H_{a-1}^{(1-\lambda )}$ of (56) also belongs to ${\mathcal {A}}$. This completes the proof. $\square$

Remark A.2

We remark that, in the case $\theta =\frac{\pi }{2}$, we can use boundary reflection to get better regularity, say global $W^{2, p}$-estimate for any p, see for example [6, Proposition 3.5].