An inverse boundary value problem for the $p$-Laplacian

TL;DR

This paper investigates a numerical approach to an inverse boundary value problem for the $p$-Laplace equation, analyzing the differentiability of the forward operator and how reconstruction accuracy depends on parameters.

Contribution

It introduces a linearization method for the inverse problem of the $p$-Laplace equation and proves the Fréchet differentiability of the forward operator for $ au > 0$, extending understanding of the problem.

Findings

Reconstruction accuracy depends on $p$ and parametrization.

The forward operator is Fréchet differentiable for $ au > 0$.

Numerical tests show the effectiveness of linearization.

Abstract

This work tackles an inverse boundary value problem for a $p$-Laplace type partial differential equation parametrized by a smoothening parameter $\tau \geq 0$. The aim is to numerically test reconstructing a conductivity type coefficient in the equation when Dirichlet boundary values of certain solutions to the corresponding Neumann problem serve as data. The numerical studies are based on a straightforward linearization of the forward map, and they demonstrate that the accuracy of such an approach depends nontrivially on $1 < p < \infty$ and the chosen parametrization for the unknown coefficient. The numerical considerations are complemented by proving that the forward operator, which maps a H\"older continuous conductivity coefficient to the solution of the Neumann problem, is Fr\'echet differentiable, excluding the degenerate case $\tau=0$ that corresponds to the classical (weighted) $p$-Laplace equation.