An inverse problem for the p-Laplacian: boundary determination

TL;DR

This paper addresses an inverse boundary value problem for the nonlinear p-Laplacian, demonstrating unique boundary determination of conductivity coefficients using nonlinear Dirichlet-to-Neumann maps with a constructive, local approach.

Contribution

It provides a novel method for boundary coefficient recovery in nonlinear elliptic equations directly from boundary measurements without linearization.

Findings

Unique determination of boundary conductivity from nonlinear boundary data

Constructive and local boundary reconstruction method

Application of complex geometrical optics solutions for complex-valued case

Abstract

We study an inverse problem for nonlinear elliptic equations modelled after the p-Laplacian. It is proved that the boundary values of a conductivity coefficient are uniquely determined from boundary measurements given by a nonlinear Dirichlet-to-Neumann map. The result is constructive and local, and gives a method for determining the coefficient at a boundary point from measurements in a small neighborhood. The proofs work with the nonlinear equation directly instead of being based on linearization. In the complex valued case we employ complex geometrical optics type solutions based on p-harmonic exponentials, while for the real case we use p-harmonic functions first introduced by Wolff.