Two uniqueness results in the inverse boundary value problem for the weighted p-Laplace equation

TL;DR

This paper establishes new uniqueness results for inverse boundary value problems involving the weighted p-Laplace equation in two and higher dimensions, using linearization techniques and conditions on weights.

Contribution

It provides the first general uniqueness theorem in the plane and extends results to higher dimensions with analytic weights under smallness conditions.

Findings

Uniqueness in the inverse problem for weighted p-Laplace in 2D with smooth weights.

Uniqueness in higher dimensions for real analytic weights with small directional derivatives.

Method based on linearizing at solutions without critical points.

Abstract

In this paper we prove a general uniqueness result in the inverse boundary value problem for the weighted p-Laplace equation in the plane, with smooth weights. We also prove a uniqueness result in dimension 3 and higher, for real analytic weights that are subject to a smallness condition on one of their directional derivatives. Both results are obtained by linearizing the equation at a solution without critical points. This unknown solution is then recovered, together with the unknown weight.