An inverse problem for semilinear elliptic equations with generalized Kerr-type nonlinearities

TL;DR

This paper extends the enclosure method to inverse problems involving semilinear elliptic equations with non-analytic nonlinearities, including Kerr-type nonlinearities, enabling shape reconstruction of unknown inclusions in nonlinear media.

Contribution

It develops an approximate solution based on linearization to adapt the enclosure method for complex nonlinearities in elliptic equations.

Findings

Applicable to a broad class of nonlinearities including Kerr and Ginzburg-Landau types

Successfully reconstructs shapes of unknown inclusions in nonlinear media

Extends existing inverse problem techniques to non-analytic nonlinear settings

Abstract

We study the inverse problem of reconstructing the shape of unknown inclusions in semilinear elliptic equations with nonanalytic nonlinearities, by extending Ikehata's enclosure method to accommodate such nonlinear effects. To address the analytical challenges, we construct an approximate solution based on the linearized equation, enabling the enclosure method to operate in this setting. The proposed method applies to a broad class of semilinear elliptic equations with non-analytic nonlinearities, including representative examples such as the Kerr-type nonlinearity, which appears in models of nonlinear optics, and the Ginzburg-Landau-type nonlinearity, which models light propagation in nonlinear dissipative media.