Extremal domains of big volume for the first eigenvalue of the Laplace-Beltrami operator in a compact manifold

International audienceWe prove the existence of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in some compact Riemannian manifolds, with volume close to the volume of the manifold. If the first (positive) eigenfunction F of the Laplace-Beltrami operator over the manifold is a nonconstant function, these domains are close to the complement of geodesic balls of small radius whose center is close to the point where F attains its maximum. If F is a constant function and the dimension of the manifold is at least 4, these domains are close to the complement of geodesic balls of small radius whose center is close to a nondegenerate critical point of the scalar curvature function

New examples of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in a Riemannian manifold with boundary

We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifold with boundary. These domains are close to half balls of small radius centered at a nondegenerate critical point of the mean curvature function of the boundary of the manifold, and their boundary intersects the boundary of the manifold orthogonally.Comment: 30 pages, 3 figure

Bifurcating extremal domains for the first eigenvalue of the Laplacian

We prove the existence of a smooth family of non-compact domains $Omega_s \subset R^{n+1}$ bifurcating from the straight cylinder $B^n \times R$ for which the first eigenfunction of the Laplacian with 0 Dirichlet boundary condition also has constant Neumann data at the boundary. The domains $Omega_s$ are rotationally symmetric and periodic with respect to the R-axis of the cylinder; they are of the form $Omega_s = {(x,t) \in R^n \times R \mid |x| < 1+s \cos((2\pi)/T_s t) + O(s^2)}$ where $T_s = T_0 + O(s)$ and T_0 is a positive real number depending on n. For $n \ge 2$ these domains provide a smooth family of counter-examples to a conjecture of Berestycki, Caffarelli and Nirenberg. We also give rather precise upper and lower bounds for the bifurcation period T_0. This work improves a recent result of the second author.Comment: 28 pages, 3 figure

Extremal domains for the first eigenvalue in a general compact Riemannian manifold

International audienceWe prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where the existence of a nondegenerate critical point of the scalar curvature of the Riemannian manifold was required

Modica type estimates and curvature results for overdetermined elliptic problems

In this paper, we establish a Modica type estimate on bounded solutions to the overdetermined elliptic problem \begin{equation*} \begin{cases} \Delta u+f(u) =0& \mbox{in $\Omega$, }\\ u>0 &\mbox{in $\Omega$, } u=0 &\mbox{on $\partial\Omega$, } \partial_{\nu} u=c_0 &\mbox{on $\partial\Omega$, } \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^{n},n\geq 2$. As we will see, the presence of the boundary changes the usual form of the Modica estimate for entire solutions. We will also discuss the equality case. From such estimates we will deduce information about the curvature of $\partial \Omega$ under a certain condition on $c_0$ and $f$. The proof uses the maximum principle together with scaling arguments and a careful passage to the limit in the arguments by contradiction.Comment: 12page

Overdetermined elliptic problems in onduloid-type domains with general nonlinearities

In this paper, we prove the existence of nontrivial unbounded domains omega subset of Rn+1, n >= 1, bifurcating from the straight cylinder BxR (where B is the unit ball of R-n), such that the overdetermined elliptic problem {delta u + f(u) = 0 in omega, u = 0 on & part;omega, & part;(nu)u = constant on & part;omega, has a positive bounded solution. We will prove such result for a very general class of functions f: [0, + infinity) -> R. Roughly speaking, we only ask that the Dirichlet problem in B admits a nondegenerate solution. The proof uses a local bifurcation argument.Spanish Government MTM2017-89677-P FQM-116China Scholarship Council CSC201906290013Ministry of Science and Innovation, Spain (MICINN) PGC2018-096422-B-I00J. Andalucia CEX2020-001105-M P18-FR-4049 A-FQM-139-UGR1

Overdetermined elliptic problems in onduloid-type domains with general nonlinearities

In this paper, we prove the existence of nontrivial unbounded domains omega subset of Rn+1, n >= 1, bifurcating from the straight cylinder BxR (where B is the unit ball of R-n), such that the overdetermined elliptic problem {delta u + f(u) = 0 in omega, u = 0 on & part;omega, & part;(nu)u = constant on & part;omega, has a positive bounded solution. We will prove such result for a very general class of functions f: [0, + infinity) -> R. Roughly speaking, we only ask that the Dirichlet problem in B admits a nondegenerate solution. The proof uses a local bifurcation argument.Spanish Government MTM2017-89677-P FQM-116China Scholarship Council CSC201906290013Ministry of Science and Innovation, Spain (MICINN) PGC2018-096422-B-I00J. Andalucia CEX2020-001105-M P18-FR-4049 A-FQM-139-UGR1