Inverse Robin Spectral Problem for the p-Laplace Operator

TL;DR

This paper studies an inverse spectral problem for the nonlinear p-Laplace operator, establishing asymptotic limits, uniqueness, and stability of recovering Robin boundary coefficients from spectral data.

Contribution

It extends classical inverse Robin problems to the nonlinear p-Laplace setting, deriving asymptotic limits, proving uniqueness, and establishing stability estimates.

Findings

Derived a p-dependent asymptotic limit for the Robin coefficient.

Proved uniqueness of the Robin coefficient from spectral data.

Established a conditional local Hölder stability estimate.

Abstract

We investigate an inverse Robin spectral problem for the $p$-Laplace operator on a bounded domain with mixed Dirichlet-Robin boundary conditions. The aim is to identify an unknown Robin coefficient on an inaccessible boundary portion from spectral information and boundary flux data measured on an accessible part. We first establish a thin-coating asymptotic limit that extends the classical result of Friedlander and Keller from the linear Laplacian to the nonlinear $p$-Laplacian. The analysis yields an effective Robin law in which the induced coefficient depends on the coating thickness through a $p$-dependent power, making explicit how the nonlinearity enters via conductivity scaling. We then prove the uniqueness of the Robin coefficient by linearizing the forward map and combining the resulting linearized equation with a boundary Cauchy unique continuation principle. Finally, we obtain a conditional local \emph{H\"older-type} stability estimate (with an explicit nonlinear remainder) by combining Fr\'echet differentiability of the solution/measurement maps with a quantitative stability bound for the \emph{linearized} inverse problem.