Conformal Geometry and The Composite Membrane Problem

TL;DR

This paper links the problem of minimizing the first Laplace eigenvalue over conformal metrics with prescribed volume to the Composite Membrane Problem, providing new insights and regularity results, and extends the framework to higher dimensions and operators.

Contribution

It establishes the equivalence between eigenvalue minimization in conformal classes and the Composite Membrane Problem, and explores higher-dimensional and higher-order free boundary problems.

Findings

Existence of a limit metric and regularity of eigenfunctions.

Complete solutions and qualitative properties derived from the Composite Membrane Problem.

Extension to higher dimensions involving GJMS and Paneitz operators.

Abstract

We consider smooth bounded surfaces with a smooth boundary and a prescribed background metric g_0. We now consider all metrics g conformal to g_0 which have a prescribed volume M. We now minimize the first eigenvalue of the Laplace operator of g over the metrics conformal to g_0 and having the prescribed volume. We show that this problem is equivalent to the study of the Composite Membrane Problem, a free boundary problem studied earlier by the author and his collaborators in all dimensions. Thus complete answers, existence of the limit metric, regularity of the minimizing eigenfunction and various qualitative properties of the metric are easily obtained from the solution of the Composite Membrane problem. The problem of minimizing eigenvalues over conformal classes has a higher dimensional analog for the critical GJMS operator and leads to new classes and questions for higher order unstable free boundary problems. In particular in dimension 4 we are lead to free boundary problems involving the Paneitz operator and in odd dimensions to fractional free boundary problems of unstable type.