On the inverse source identification problem in $L^\infty$ for fully nonlinear elliptic PDE

TL;DR

This paper extends previous work on inverse source problems by addressing the fully nonlinear elliptic PDE case, introducing a novel regularisation method to handle nonlinearity and ill-posedness in an $L^ Infty$ setting.

Contribution

It develops a two-parameter Tykhonov regularisation with a higher order $L^2$ viscosity term for fully nonlinear elliptic equations, improving stability and approximation.

Findings

Introduces a new regularisation strategy for nonlinear inverse problems.

Proves convergence of the regularisation method under certain conditions.

Handles nonconvexity and lack of weak continuity in the problem.

Abstract

In this paper we generalise the results proved in [N. Katzourakis, An $L^\infty$ regularisation strategy to the inverse source identification problem for elliptic equations, SIAM J. Math. Anal. 51:2, 1349-1370 (2019)] by studying the ill-posed problem of identifying the source of a fully nonlinear elliptic equation. We assume Dirichlet data and some partial noisy information for the solution on a compact set through a fully nonlinear observation operator. We deal with the highly nonlinear nonconvex nature of the problem and the lack of weak continuity by introducing a two-parameter Tykhonov regularisation with a higher order $L^2$ "viscosity term" for the $L^\infty$ minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals.