Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A characterization of capillary spherical caps by a partially overdetermined problem in a half ball

Xiaohan Jia, Zheng Lu, Chao Xia, Xuwen Zhang

TL;DR

The paper characterizes capillary spherical caps in a half-ball via a Serrin-type partially overdetermined boundary value problem. It develops an integral identity framework around a P-function and an auxiliary quadratic function to derive a rigidity result: if a solution with f ≤ 0 exists under the mixed boundary conditions, then the domain is a spherical cap and f is quadratic, with the contact angle constrained by cos θ = - c / c0. This yields a complete classification for general contact angles θ ∈ (0, π) and connects Serrin-type overdetermined problems with capillary constant-mean-curvature geometry. The work extends half-space results to the half-ball, highlighting the interplay between boundary conditions, geometry of capillary surfaces, and rigidity phenomena.

Abstract

In this note, we study a Serrin-type partially overdetermined problem proposed by Guo-Xia (Calc. Var. Partial Differential Equations 58: no. 160, 2019. https://doi.org/10.1007/s00526-019-1603-3, and prove a rigidity result that characterizes capillary spherical caps in a half ball.

A characterization of capillary spherical caps by a partially overdetermined problem in a half ball

TL;DR

The paper characterizes capillary spherical caps in a half-ball via a Serrin-type partially overdetermined boundary value problem. It develops an integral identity framework around a P-function and an auxiliary quadratic function to derive a rigidity result: if a solution with f ≤ 0 exists under the mixed boundary conditions, then the domain is a spherical cap and f is quadratic, with the contact angle constrained by cos θ = - c / c0. This yields a complete classification for general contact angles θ ∈ (0, π) and connects Serrin-type overdetermined problems with capillary constant-mean-curvature geometry. The work extends half-space results to the half-ball, highlighting the interplay between boundary conditions, geometry of capillary surfaces, and rigidity phenomena.

Abstract

In this note, we study a Serrin-type partially overdetermined problem proposed by Guo-Xia (Calc. Var. Partial Differential Equations 58: no. 160, 2019. https://doi.org/10.1007/s00526-019-1603-3, and prove a rigidity result that characterizes capillary spherical caps in a half ball.
Paper Structure (5 sections, 3 theorems, 45 equations)

This paper contains 5 sections, 3 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Notations
  4. Preliminaries
  5. Proof of Theorem \ref{['Thm-rigid-BVP-halfball']}

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{B}_{+}^{n+1}$ be a connected open set, whose boundary $\partial\Omega$ is made of two smooth parts ${\Sigma}\coloneqq{\partial \Omega\cap \mathbb{B}_{+}^{n+1}}$, $T\coloneqq\partial\Omega\setminus\overline{\Sigma}$. If the mixed boundary problem $(eq-mixed-halfball)$, with where $z\in\mathbb{R}^{n+1}$ satisfies while the solution $f$ is given by In particular, $\Sigma$

Theorems & Definitions (7)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • proof : Proof of Theorem \ref{['Thm-rigid-BVP-halfball']}