Determining an unbounded potential for an elliptic equation with a power type nonlinearity

TL;DR

This paper demonstrates that potentials in certain Lebesgue spaces can be uniquely determined from full or partial Dirichlet-to-Neumann maps in semilinear elliptic equations, extending previous results to less regular potentials.

Contribution

It extends inverse problem results to potentials in $L^{n/2+ ext{epsilon}}$ spaces, broadening the class of potentials recoverable from boundary measurements.

Findings

Unique determination of potentials in $L^{n/2+ ext{epsilon}}$ from full and partial data.

Potential recovery in $L^{n+ ext{epsilon}}$ from data at a single boundary point.

Extension of previous results from H"older continuous to Lebesgue space potentials.

Abstract

In this article we focus on inverse problems for a semilinear elliptic equation. We show that a potential $q$ in $L^{n/2+\varepsilon}$, $\varepsilon>0$, can be determined from the full and partial Dirichlet-to-Neumann map. This extends the results from [M. Lassas, T. Liimatainen, Y.-H. Lin, and M. Salo, Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations, Rev. Mat. Iberoam. (2021)] where this is shown for H\"older continuous potentials. Also we show that when the Dirichlet-to-Neumann map is restricted to one point on the boundary, it is possible to determine a potential $q$ in $L^{n+\varepsilon}$. The authors of arXiv:2202.05290 [math.AP] proved this to be true for H\"older continuous potentials.