Geometry and topology of some overdetermined elliptic problems

Abstract

International audienceWe prove some geometric and topological properties for unbounded domains of the plane that support a positive solution to some elliptic equations, with 0 Dirichlet and constant Neumann boundary condition. Some of such properties are true also in higher dimension. Such properties give a partial answer to a conjecture of Berestycki-Caffarelli-Nirenberg in dimension 2

A Rigidity Result for Overdetermined Elliptic Problems in the Plane

“…Here ν(x) stands for the interior normal vector to ∂Ω at x. In this case we say that Ω is an f -extremal domain (see [28] for a motivation of that definition). The case of bounded f -extremal domains was completely solved by J. Serrin in [29] (see also [26]): the ball is the unique such domain and any solution is radial.…”

“…In [36] the case of f -extremal epigraphs is solved for some nonlinearities f of the Allen-Cahn type. Finally, in [28] the result is proved if either f (t) ≥ λt or Ω is contained in a half-plane and ∇u is bounded (see also [13] for a generalization to other geometries). Observe that the assumption f (t) ≥ t excludes the prototypical Allen-Cahn nonlinearity; we point out that not even the half-plane is an f -extremal domain for those nonlinearities f .…”

Extremal domains on Hadamard manifolds

“…A. Ros and P. Sicbaldi [35] proved narrow properties for f -extremal domains in the Euclidean Space based on geometric ideas developed in [15] for CMC surfaces. We are able to extend these geometric ideas to the context of OEP in Hadamard manifolds.…”

“…More precisely, they proved that if Ω is contained in a half-plane and |∇u| is bounded, or if there exists a positive constant λ such that f (t) λ t for all t > 0, then the BCN-Conjecture is true for n = 2. Besides, A. Ros and P. Sicbaldi [35] have also shown that some classical results in the theory of CMC hypersurfaces extend to the context of OEPs (see [35, Theorems 2.2, 2.8 and 2.13]). From the above discussion, we know that the OEP is an interesting and important topic, which is worthy of investigating and still has some unsolved problems left.…”

“…Remark 5.2. By [35,Remark 5.2] and the method in the proof of Proposition 5.1, it is easy to explain that the Neumann boundary data α cannot vanish.…”