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

TL;DR

This work analyzes the Robin $p$-Laplacian first eigenvalue under domain perturbations using Hadamard-type variational methods. It derives two equivalent shape derivative formulas for $\dot{\lambda}_1(0)$ that involve boundary integrals with $|\nabla u|^p$, $|u|^p$, mean curvature $H$, and the perturbation field $v$, addressing regularity via a regularization approach. The paper also establishes domain monotonicity results, proving a ball-case criterion and showing that, for sufficiently large boundary parameter $\beta$, domain monotonicity persists for all smooth domains through a Dirichlet-limit convergence $\lambda_1^\beta\to\lambda_1^D$ and $\phi_\beta\to\phi_D$ in $C^1(\overline{\Omega})$. These findings advance understanding of spectral shape optimization for nonlinear Robin problems and provide tools for comparing domains in Robin $p$-Laplacian contexts.

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.

Theorems & Definitions (4)

• proof
• proof
• proof : Proof of Theorem \ref{['Ball case']}
• proof