On a shape derivative formula for the Robin $p$-Laplace eigenvalue

TL;DR

This paper derives a shape derivative formula for the first eigenvalue of the Robin p-Laplace operator and investigates how it varies with domain perturbations, especially for large boundary parameters.

Contribution

It provides the first shape derivative formula for the Robin p-Laplace eigenvalue and analyzes its monotonicity under domain inclusion for large boundary parameters.

Findings

Shape derivative formula for the first Robin p-Laplace eigenvalue.

Monotonicity of the eigenvalue with respect to domain inclusion for large boundary parameters.

Insight into eigenvalue variation under domain perturbations.

Abstract

We obtain shape derivative formulae for the first eigenvalue of the Robin $p$-Laplace operator. This result is used to study the variation of the first eigenvalue with respect to perturbations of the domain. In particular, we prove that for large values of the boundary parameter, the first eigenvalue is monotonic with respect to domain inclusion for smooth domains.