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… 
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References

SHOWING 1-10 OF 35 REFERENCES
New extremal domains for the first eigenvalue of the Laplacian in flat tori
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
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
Stability of Hypersurfaces of Constant Mean Curvature in Riemannian Manifolds.
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
On De Giorgi's conjecture in dimension N>9
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
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
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
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
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
...
1
2
3
4
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