Enclosure method for the p-Laplace equation

TL;DR

This paper extends the enclosure method to the nonlinear p-Laplace equation, enabling the reconstruction of the convex hull of an inclusion using special solutions, with applications to inverse problems in nonlinear conductivity.

Contribution

It introduces a novel approach for the p-Calderón problem, generalizing the inverse conductivity problem to nonlinear settings with theoretical justification.

Findings

Successfully reconstructs convex hulls of inclusions in nonlinear models

Establishes a monotonicity inequality for the p-Laplace equation

Validates the method for penetrable obstacles with jump conductivity

Abstract

We study the enclosure method for the p-Calder\'on problem, which is a nonlinear generalization of the inverse conductivity problem due to Calder\'on that involves the p-Laplace equation. The method allows one to reconstruct the convex hull of an inclusion in the nonlinear model by using exponentially growing solutions introduced by Wolff. We justify this method for the penetrable obstacle case, where the inclusion is modelled as a jump in the conductivity. The result is based on a monotonicity inequality and the properties of the Wolff solutions.