Inverse problems for semilinear elliptic equations with low regularity

TL;DR

This paper proves the unique determination of a general nonlinearity in semilinear elliptic equations from boundary data, extending previous results to low regularity settings.

Contribution

It provides low regularity versions of inverse problem results for semilinear elliptic equations, allowing for less smooth nonlinearities.

Findings

Nonlinearity $a(x,u)$ is uniquely determined from boundary measurements.

Results hold even with low regularity assumptions on the nonlinearity.

Extends previous high regularity results to more general, less smooth cases.

Abstract

We show that a general nonlinearity $a(x,u)$ is uniquely determined, possibly up to a gauge, in a neighborhood of a fixed solution from boundary measurements of the corresponding semilinear equation. The main theorems are low regularity counterparts of the results in our recent paper (Johansson, Nurminen, Salo; ArXiv preprint 2312.12196).