# 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

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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
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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
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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] …

A rigidity result for overdetermined elliptic problems in the plane

- Mathematics
- 2015

Let $f:[0,+\infty) \to \mathbb{R}$ be a (locally) Lipschitz function and $\Omega \subset \mathbb{R}^2$ a $C^{1,\alpha}$ domain whose boundary is unbounded and connected. If there exists a positive…

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…

Nontrivial solutions to Serrin's problem in annular domains

- Physics, Mathematics
- 2019

We construct nontrivial smooth bounded domains $\Omega \subseteq \mathbb{R}^n$ of the form $\Omega_0 \setminus \overline{\Omega}_1$, bifurcating from annuli, for which there exists a positive…

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…

## References

SHOWING 1-10 OF 35 REFERENCES

New extremal domains for the first eigenvalue of the Laplacian in flat tori

- Mathematics
- 2010

We prove the existence of nontrivial compact extremal domains for the first eigenvalue of the Laplacian in manifolds $${\mathbb{R}^{n}\times \mathbb{R}{/}T\, \mathbb{Z}}$$ with flat metric, for some…

Constant mean curvature surfaces with Delaunay ends

- Mathematics
- 1998

In this paper we shall present a construction of complete surfaces M in R3 with finitely many ends and finite topology, and with nonzero constant mean curvature (CMC). This construction is parallel…

Geometry and Topology of some overdetermined elliptic problems

- Mathematics
- 2012

We study necessary conditions on the geometry and the topology of domains in $\mathbb{R}^2$ that support a positive solution to a classical overdetermined elliptic problem. The ideas and tools we use…

Stability of Hypersurfaces of Constant Mean Curvature in Riemannian Manifolds.

- Mathematics
- 1988

Hypersurfaces \(M^n\)with constant mean curvature in a Riemannian manifold \(\overline{M}^{n+1}\)display many similarities with minimal hypersurfaces of \(\overline{M}^{n+1}\). They are both…

BIFURCATING EXTREMAL DOMAINS FOR THE FIRST EIGENVALUE OF THE LAPLACIAN

- Mathematics
- 2011

Abstract We prove the existence of a smooth family of non-compact domains Ω s ⊂ R n + 1 , n ⩾ 1 , bifurcating from the straight cylinder B n × R for which the first eigenfunction of the Laplacian…

On De Giorgi's conjecture in dimension N>9

- Mathematics
- 2011

A celebrated conjecture due to De Giorgi states that any bounded solution of the equation u + (1 u 2 )u = 0 in R N with @yNu > 0 must be such that its level setsfu = g are all hyperplanes, at least…

Entire solutions of semilinear elliptic equations in R^3 and a conjecture of De Giorgi

- Mathematics
- 2000

This paper is concerned with the study of bounded solutions of semilinear elliptic equations u F u in the whole space R under the assumption that u is monotone in one direction say nu in R n The goal…

Regularity of flat level sets in phase transitions

- Mathematics
- 2009

We consider local minimizers of the Ginzburg-Landau energy functional ∫1/2|∇u| 2 +1/4(1-u 2 ) 2 dx and prove that, if the 0 level set is included in a flat cylinder then, in the interior, it is…

1D symmetry for solutions of semilinear and quasilinear elliptic equations

- Mathematics
- 2011

Several new $ 1$D results for solutions of possibly singular or degenerate elliptic equations, inspired by a conjecture of De Giorgi, are provided. In particular, $ 1$D symmetry is proven under the…

Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds

- Mathematics
- 2012

Our work proposes a unified approach to three different topics in a general Riemannian setting: splitting theorems, symmetry results and overdetermined elliptic problems. By the existence of a stable…