Partial data inverse problems for quasilinear conductivity equations

TL;DR

This paper proves that partial boundary measurements uniquely determine nonlinear conductivities in quasilinear equations, using a novel $L^1$-density approach involving harmonic functions.

Contribution

It establishes unique identifiability of nonlinear conductivities from partial boundary data for quasilinear equations, extending previous results to partial boundary measurements.

Findings

Unique determination of nonlinear conductivities from partial boundary data.

Development of an $L^1$-density technique involving harmonic functions.

Applicability to classes of semilinear and quasilinear conductivity equations.

Abstract

We show that the knowledge of the Dirichlet-to-Neumann maps given on an arbitrary open non-empty portion of the boundary of a smooth domain in $\mathbb{R}^n$, $n\ge 2$, for classes of semilinear and quasilinear conductivity equations, determines the nonlinear conductivities uniquely. The main ingredient in the proof is a certain $L^1$-density result involving sums of products of gradients of harmonic functions which vanish on a closed proper subset of the boundary.