Identifiability of Diffusion Coefficients for Source Terms of Non-Uniform Sign

TL;DR

This paper investigates the conditions under which a diffusion coefficient in an elliptic PDE can be uniquely identified from solutions, especially when the source term changes sign, providing new insights into inverse problems with minimal regularity assumptions.

Contribution

It establishes identifiability results for diffusion coefficients with sign-changing source terms under mild regularity conditions, including detailed analysis for one-dimensional domains.

Findings

Identifiability of diffusion coefficient $a$ from solution $u$ under sign-changing source $f$.

Conditions ensuring the gradient of $u$ is nonzero almost everywhere.

Continuity properties of the mapping from $u$ to $a$ in one-dimensional cases.

Abstract

The problem of recovering a diffusion coefficient $a$ in a second-order elliptic partial differential equation from a corresponding solution $u$ for a given right-hand side $f$ is considered, with particular focus on the case where $f$ is allowed to take both positive and negative values. Identifiability of $a$ from $u$ is shown under mild smoothness requirements on $a$, $f$, and on the spatial domain $D$, assuming that either the gradient of $u$ is nonzero almost everywhere, or that $f$ as a distribution does not vanish on any open subset of $D$. Further results of this type under essentially minimal regularity conditions are obtained for the case of $D$ being an interval, including detailed information on the continuity properties of the mapping from $u$ to $a$.