A partially overdetermined problem in domains with partial umbilical boundary in space forms

通过半圆球中的部分超定问题刻画毛细球冠

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

本文研究Guo-Xia提出的Serrin型部分超定问题，证明了一个刚性结果，刻画了半圆球中的毛细球冠。

On Serrin's overdetermined problem and a conjecture of Berestycki, Caffarelli and Nirenberg

This paper concerns rigidity results to Serrin's overdetermined problem in an epigraph $$ \{\begin{aligned} &Δu+ f(u)=0,\ \ \ {in}\ Ω=\{(x^\prime,x_n): x_n>φ(x^\prime)\},\\ &u>0,\ \ \ {in}\ Ω,\\ &u=0,\ \ \ {on}\ \partialΩ,\\ &|\nabla u|=const. {on} \partialΩ. \end{aligned}. $$ We prove that up to isometry the epigraph must be an half space and that the solution $u$ must be one-dimensional, provided that one of the following assumptions are satisfied: either $n=2$; or $ φ$ is globally Lipschitz, or $n \leq 8$ and $ \frac{\partial u}{\partial x_n} >0$ in $Ω$. In view of the counterexample constr…

An exterior overdetermined problem for Finsler $N$-laplacian in convex cones

We consider a partially overdetermined problem for anisotropic $N$-Laplace equations in a convex cone $Σ$ intersected with the exterior of a bounded domain $Ω$ in $\mathbb{R}^N$, $N\geq 2$. Under a prescribed logarithmic condition at infinity, we prove a rigidity result by showing that the existence of a solution implies that $Σ\capΩ$ must be the intersection of the Wulff shape and $Σ$. Our approach is based on a Pohozaev-type identity and the characterization of minimizers of the anisotropic isoperimetric inequality inside convex cones.

Overdetermined problems and relative Cheeger sets in unbounded domains

In this paper we study a partially overdetermined mixed boundary value problem for domains $Ω$ contained in an unbounded set $\mathcal C$. We introduce the notion of Cheeger set relative to $\mathcal C$ and show that if a domain $Ω\subset \mathcal C$ admits a solution of the overdetermined problem, then it coincides with its relative Cheeger set. We also study the related problem of characterizing constant mean curvature surfaces $Γ$ inside $\mathcal C$. In the case when $\mathcal C$ is a cylinder we obtain further results whenever the relative boundary of $Ω$ or the surface $Γ$ is a graph on …

从任意高特征值出发的过定椭圆问题的分支域

Nonsymmetric sign-changing solutions for the partially overdetermined eigenvalue problem on a hollow cylinder

研究了无界域上过定特征值问题，构造了非平凡无界域家族，揭示了高特征值问题的复杂性，并为Berenstein猜想提供了反例。

无界周期域中具有变号特征函数的超定问题

Overdetermined problems with sign-changing eigenfunctions in unbounded periodic domains

本文证明欧几里得空间中存在非平凡无界周期径向域，使 Laplace Dirichlet 特征函数可变号且边界 Neumann 值恒定，并推广到双边界分量的相反边界值情形。

Overdetermined problems and constant mean curvature surfaces in cones

We consider a partially overdetermined problem in a sector-like domain $Ω$ in a cone $Σ$ in $\mathbb{R}^N$, $N\geq 2$, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that $Ω$ is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces $Γ$ with boundary which satisfy a "gluing" condition with respect to the cone $Σ$. We prove that if either the cone is convex or the surface is a radial graph then $Γ$ must be a spherical cap. Finally we show that, …

关于上图像的Serrin超定问题

Serrin's overdetermined problems on epigraphs

研究Serrin超定问题，当Ω为上图像且u为经典解时，在f为特定类型及维度条件下，证明Ω是仿射半空间、u是一维的，部分回答相关问题，且当f(0)<0时在任意N≥2维度有新单调性结果。

关于均值性质与若干加权Poincaré型不等式的注记

Remarks about the mean value property and some weighted Poincaré-type inequalities

本文针对穿孔区域调和函数超定问题给出定量稳定性定理，提出向量场新加权Poincaré型不等式，还得到锥内调和函数均值性质及相关不等式并推广了经典结果。