An inverse boundary value problem for certain anisotropic quasilinear elliptic equations

TL;DR

This paper establishes the uniqueness of solutions in an inverse boundary value problem for certain anisotropic quasilinear elliptic equations, expanding the class of nonlinearities and anisotropic coefficients that can be uniquely identified.

Contribution

It introduces a novel approach allowing anisotropic coefficients in the nonlinear part of quasilinear elliptic equations, using Gaussian quasi-modes to prove uniqueness.

Findings

Proved uniqueness for anisotropic nonlinear coefficients.

Developed a construction method using Gaussian quasi-modes.

Extended inverse boundary value problem results to anisotropic cases.

Abstract

In this paper we prove uniqueness in the inverse boundary value problem for quasilinear elliptic equations whose linear part is the Laplacian and nonlinear part is the divergence of a function analytic in the gradient of the solution. The main novelty in terms of the result is that the coefficients of the nonlinearity are allowed to be "anisotropic". As in previous works, the proof reduces to an integral identity involving the tensor product of the gradients of 3 or more harmonic functions. Employing a construction method using Gaussian quasi-modes, we obtain a convenient family of harmonic functions to plug into the integral identity and establish our result.