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

Shortest closed billiard orbits on convex tables

Given a planar compact convex billiard table $T$, we give an algorithm to find the shortest generalised closed billiard orbits on $T$. (Generalised billiard orbits are usual billiard orbits if $T$ has smooth boundary.) This algorithm is finite if $T$ is a polygon and provides an approximation scheme in general. As an illustration, we show that the shortest generalised closed billiard orbit in a regular $n$-gon $R_n$ is 2-bounce for $n \ge 4$, with length twice the width of $R_n$. As an application we obtain an algorithm computing the Ekeland-Hofer-Zehnder capacity of the four-dimensional domain $T \times B^2$ in the standard symplectic vector space $\mathbb{R}^4$. Our method is based on the work of Bezdek-Bezdek and on the uniqueness of the Fagnano triangle in acute triangles. It works, more generally, for planar Minkowski billiards.Comment: 16 pages, 11 figure