An inverse problem for semilinear elliptic equations with generalized Kerr-type nonlinearities
Pu-Zhao Kow, Rulin Kuan
TL;DR
The paper addresses the inverse boundary value problem for semilinear elliptic equations with non-analytic Kerr-type nonlinearities, aiming to reconstruct the convex hull of an unknown inclusion from boundary measurements. It extends Ikehata's enclosure method by constructing approximate solutions from the linearized problem and introducing an indicator functional that can be approximated linearly, with complex geometrical optics test data guiding the reconstruction. A key contribution is showing that the convex hull of the inclusion is uniquely determined by the Dirichlet-to-Neumann map under a broad class of nonlinearities, including Kerr-type and Ginzburg-Landau-type forms, and providing a rigorous mechanism to recover the inclusion via the asymptotic behavior of the indicator. The work has potential implications for imaging in nonlinear optical and dissipative media, offering a robust, boundary-data-driven tool for nondestructive evaluation in settings with non-analytic responses.
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.