Prescribed nonlinearity helps in an anisotropic Calder\'on-type problem

TL;DR

This paper proves that for a specific class of nonlinear anisotropic elliptic equations, the boundary data uniquely determine the coefficient matrix in dimensions three and higher, overcoming classical obstructions present in linear cases.

Contribution

It establishes uniqueness results for an inverse boundary value problem involving a quasilinear, anisotropic elliptic equation with prescribed nonlinearity, extending Calderón-type results.

Findings

Boundary data determine the coefficient matrix uniquely in dimensions 3 and higher.

Contrasts with classical linear Calderón problem where uniqueness fails due to invariance.

Shows prescribed nonlinearity aids in solving the inverse problem.

Abstract

In this paper I consider the inverse boundary value problem for a quasilinear, anisotropic, elliptic equation of the form $\nabla\cdot(\gamma\nabla u+|\nabla u|^{p-2}\nabla u)=0$, where $\gamma$ is a smooth, matrix valued, function with a uniform lower bound. I show that boundary Dirichlet and Neumann data for this equation, in the form of a Dirichlet-to-Neumann map, determine the coefficient matrix uniquely, in dimension 3 and higher. This stands in contrast to the classical linear anisotropic Calder\'on problem where there is a known obstruction to uniqueness due to the invariance of the boundary data under transformations of the equation via any boundary fixing diffeomorphism.