Mini-Workshop: Shape Analysis for Eigenvalues

Abstract

The main goal of the meeting was to bring together two mathematical communities working on the shape analysis of eigenvalues by quite different methods. On the one hand explicit solutions to optimal shape problems for eigenvalues are searched by means of direct estimations, symmetrizations, rearrangements; on the other hand recent techniques of variational type have been developed to prove the existence of an optimal shape and intensive research is carried out to prove the regularity of the free boundary and to analyze the optimality Mathematics Subject Classification (2000): 49Q10, 35P15, 49R50. Introduction by the Organisers The mini workshop Shape analysis for eigenvalues, organised by Dorin Bucur (Chambery), Giuseppe Buttazzo (Pisa) and Antoine Henrot (Nancy) was held April 8th–April 14th, 2007. This meeting was attended by 18 participants. The question of localizing or optimizing the eigenvalues differential operators has applications in several domains like acoustics, quantum mechanics, visualization, solid or fluid mechanics and bio-mathematics. Those questions have also a specific mathematical interest since they melt geometrical questions with analysis of partial differential equations and provide model problems for general and applied shape optimization problems. Despite (or because) their false simplicity, several problems are still open, although formulated hundreds year ago (see for instance the recent books by Bucur & Buttazzo Variational Methods in Shape Optimization Problems, BirkhauserVerlag, 2005 and Henrot Extremum problems for eigenvalues of elliptic operators, Birkhauser, 2006) and give rise to important debates within the international scientific community. Among them, we can cite 996 Oberwolfach Report 18/2007 minimization of the eigenvalues of the Dirichlet Laplacian maximization of the eigenvalues of the Neumann Laplacian estimates of the fundamental gap for the Laplace or Schrodinger operators the hot spot or the nodal line conjecture minimization of the buckling load Each of these problems is associated to one or several constraints for the geometries, which may be local or not local (convexity, connectedness). One of the goals of the proposed mini-workshop was to bring together two mathematical communities working on the topics above by quite different methods. On the one hand explicit solutions to optimal shape problems for eigenvalues are searched by means of direct estimations, symmetrizations, rearrangements; on the other hand recent techniques of variational type have been developed to prove the existence of an optimal shape and intensive research is carried out to prove the regularity of the free boundary and to analyze the optimality conditions. A second main issue was to present the state of the art on some famous open problems and conjectures, and try to make a step forward in the direction of solving them. The organization of the mini-workshop was the following: mornings were devoted to lectures by participants while each afternoon began by an open problems session and smaller working groups (4-5 persons) were organized to work on these problems.