Towards a reversed Faber-Krahn inequality for the truncated Laplacian

TL;DR

This paper investigates a nonlinear eigenvalue problem for degenerate elliptic operators, demonstrating that, unlike the Laplacian, symmetric square domains maximize the principal eigenvalue among fixed-volume domains.

Contribution

It introduces a reversed Faber-Krahn inequality for a class of degenerate elliptic operators, showing symmetry maximizes the principal eigenvalue for certain domains.

Findings

Symmetry of square domains maximizes the principal eigenvalue.

Contrasts with classical Laplacian behavior where symmetry minimizes the eigenvalue.

Provides new insights into eigenvalue optimization for degenerate operators.

Abstract

We consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for a class of very degenerate elliptic operators, with the aim to show that, at least for square type domains having fixed volume, the symmetry of the domain maximize the principal eigenvalue, contrary to what happens for the Laplacian.