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Inverse Robin Spectral Problem for the p-Laplace Operator

Farid Bozorgnia, Olimjon Eshkobilov

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{Hölder-type} stability estimate (with an explicit nonlinear remainder) by combining Fréchet differentiability of the solution/measurement maps with a quantitative stability bound for the \emph{linearized} inverse problem.

Inverse Robin Spectral Problem for the p-Laplace Operator

Abstract

We investigate an inverse Robin spectral problem for the -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 -Laplacian. The analysis yields an effective Robin law in which the induced coefficient depends on the coating thickness through a -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{Hölder-type} stability estimate (with an explicit nonlinear remainder) by combining Fréchet differentiability of the solution/measurement maps with a quantitative stability bound for the \emph{linearized} inverse problem.
Paper Structure (17 sections, 19 theorems, 184 equations, 1 figure)

This paper contains 17 sections, 19 theorems, 184 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. Background and Preliminaries
  4. The $p$-Laplacian and Principal Eigenvalue Problem
  5. Thin Coating Asymptotics and Effective Robin Conditions
  6. Limiting Cases; $p \to 1^+$ and $p \to \infty$
  7. Inverse Problem: Uniqueness and Stability
  8. Boundary Non-Degeneracy and Localized Regularization
  9. Uniqueness
  10. Fréchet Differentiability of the Solution Map
  11. Stability Estimate
  12. Verification of Assumptions
  13. Boundary non-degeneracy and the collar $\mathcal{U}_\eta$.
  14. Assumption \ref{['ass:path_ellipticity']}:
  15. Assumption \ref{['ass:lin_inv']} (invertibility of the augmented linearization).
  16. ...and 2 more sections

Key Result

Proposition 2.2

Let $\Omega\subset\mathbb R^n$ be bounded with $C^2$ boundary, let $\partial\Omega=\Gamma_D\cup\gamma$ with $\Gamma_D$ and $\gamma$ disjoint relatively open subsets of $\partial\Omega$ of positive surface measure, and let $h\in L^\infty(\gamma)$ with $h\ge 0$. Define $\lambda_1(h)$ by the Rayleigh q

Figures (1)

  • Figure 1: The inner domain $\Omega \subset \mathbb{R}^n$ (blue) is surrounded by a thin coating layer (orange annulus) of variable thickness $\varepsilon\rho(\xi)$, forming the extended domain $\Omega_\varepsilon = \Omega \cup \Sigma_\varepsilon$. The conductivity is $\sigma = 1$ in $\Omega$ and $\sigma = \varepsilon^{p-1}$ in the coating $\Sigma_\varepsilon$. The choice of the exponent $(p-1)$ ensures that the effective boundary condition inherits the correct scaling behavior of the nonlinear $p$-Laplacian.

Theorems & Definitions (42)

  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • ...and 32 more