# Serrin’s overdetermined problem and constant mean curvature surfaces

@article{Pino2015SerrinsOP, title={Serrin’s overdetermined problem and constant mean curvature surfaces}, author={Manuel del Pino and Frank Pacard and Juncheng Wei}, journal={Duke Mathematical Journal}, year={2015}, volume={164}, pages={2643-2722} }

For all $N \geq 9$, we find smooth entire epigraphs in $\R^N$, namely smooth domains of the form $\Omega : = \{x\in \R^N\ / \ x_N > F (x_1,\ldots, x_{N-1})\}$, which are not half-spaces and in which a problem of the form
$\Delta u + f(u) = 0 $ in $\Omega$ has a positive, bounded solution with 0 Dirichlet boundary data and constant Neumann boundary data on $\partial \Omega$. This answers negatively for large dimensions a question by Berestycki, Caffarelli and Nirenberg \cite{bcn2}. In 1971…

## 24 Citations

Serrin’s overdetermined problem on the sphere

- Mathematics
- 2016

We study Serrin’s overdetermined boundary value problem $$\begin{aligned} -\Delta _{S^N}\, u=1 \quad \text { in }\Omega ,\quad u=0, \; \partial _\eta u={\text {const}} \quad \text {on }\partial…

Unbounded Periodic Solutions to Serrin’s Overdetermined Boundary Value Problem

- Mathematics
- 2016

AbstractWe study the existence of nontrivial unbounded domains $${\Omega}$$Ω in $${{\mathbb R}^{N}}$$RN such that the overdetermined problem
$${-\Delta u = 1 \quad {\rm in} \, \Omega}, \quad u = 0,…

Solutions to overdetermined elliptic problems in nontrivial exterior domains

- MathematicsJournal of the European Mathematical Society
- 2019

In this paper we construct nontrivial exterior domains $\Omega \subset \mathbb{R}^N$, for all $N\geq 2$, such that the problem $$\left\{ {ll} -\Delta u +u -u^p=0,\ u >0 & \mbox{in }\; \Omega, {1mm] …

Serrin's over-determined problem on Riemannian manifolds

- Mathematics
- 2014

Abstract Let (ℳ,g)${(\mathcal {M},g)}$ be a compact Riemannian manifold of dimension N, N ≥ 2. In this paper, we prove that there exists a family of domains (Ω ε ) ε∈(0,ε 0 ) ${(\Omega _\varepsilon…

Serrin’s overdetermined problem for fully
nonlinear nonelliptic equations

- MathematicsAnalysis & PDE
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Let $u$ denote a solution to a rotationally invariant Hessian equation $F(D^2u)=0$ on a bounded simply connected domain $\Omega\subset R^2$, with constant Dirichlet and Neumann data on $\partial…

Geometric rigidity of constant heat flow

- MathematicsCalculus of Variations and Partial Differential Equations
- 2018

Let $$\Omega $$Ω be a compact Riemannian manifold with smooth boundary and let $$u_t$$ut be the solution of the heat equation on $$\Omega $$Ω, having constant unit initial data $$u_0=1$$u0=1 and…

On Smooth Solutions to One Phase-Free Boundary Problem in $\mathbb{R}^{n}$

- MathematicsInternational Mathematics Research Notices
- 2019

We construct a smooth axially symmetric solution to the classical one phase free boundary problem in $\mathbb{R}^{n}$, $n\geq 3.$ Its free boundary is of “catenoid” type. This is a higher…

On one phase free boundary problem in $\mathbb{R}^{N}$

- Mathematics
- 2017

We construct a smooth axially symmetric solution to the classical one phase free boundary problem in $\mathbb{R}^{N}$. Its free boundary is of \textquotedblleft catenoid\textquotedblright\ type. This…

A Rigidity Result for Overdetermined Elliptic Problems in the Plane

- Mathematics
- 2015

Let f:[0,+∞)→ℝ be a (locally) Lipschitz function and Ω⊂ℝ2 a C1,α domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problem…

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