International audienceWe prove the existence of nontrivial and noncompact extremal domains for the first eigenvalue of the Laplacian in some flat tori. Such domains can be extended by periodicity to nontrivial and noncompact domains in Euclidean spaces whose first eigenfunction of the Laplacian with 0 Dirichlet boundary condition has also constant Neumann data at the boundary, providing a couterexemple to a conjecture of Berestycki-Caffarelli-Nirenberg in dimension bigger or equal then 3. These domains are close to a straigh cylinder, they are invariant by rotation with respect to the vertical axe, and are not invariant by vertical translations
International audienceWe prove the existence of a smooth family of noncompact domains of the euclidean space where the first eigenfunction of the Laplacian with 0 Dirichlet boundary condition has also constant Neumann data at the boudary. These domains are rotationally symmetric and periodic with respect to a vertical axe. We determine also the shape of these domains, and precise upper and lower bounds for their period. These domains provide a smooth family of counterexemples to a conjecture of Berestycki-Caffarelli-Nirenberg in dimension bigger or equal then 3
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
ABSTRACT. Let f : [0, +∞) → R be a (locally) Lipschitz function and Ω ⊂ R 2 a C 1,α domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problemwe prove that Ω is a half-plane. In particular, we obtain a partial answer to a question raised by H. Berestycki, L. Caffarelli and L. Nirenberg in 1997.
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