Eigenvalues of the Finsler $p$-Laplacian on varying domains

TL;DR

This paper investigates how the first eigenvalue and eigenfunctions of the Finsler p-Laplacian change with domain perturbations, extending classical results and deriving new formulas and identities.

Contribution

It generalizes known results for the standard p-Laplacian to the Finsler setting, including differentiability, Hadamard formulas, and continuity of eigenfunctions.

Findings

Proved Frechét differentiability of eigenvalues.

Derived Hadamard formulas for eigenvalues.

Established continuity of eigenfunctions.

Abstract

We study the dependence of the first eigenvalue of the Finsler $p$-Laplacian and the corresponding eigenfunctions upon perturbation of the domain and we generalize a few results known for the standard $p$-Laplacian. In particular, we prove a Frech\'{e}t differentiability result for the eigenvalues, we compute the corresponding Hadamard formulas and we prove a continuity result for the eigenfunctions. Finally, we briefly discuss a well-known overdetermined problem and we show how to deduce the Rellich-Pohozaev identity for the Finsler $p$-Laplacian from the Hadamard formula.