arXiv subjects

Rulin Kuan

Publications and source records attributed to Rulin Kuan.

5 recordsLinked to original sources

arXiv2025

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

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.

math.AP↗

arXiv2023

The encloure method for semilinear elliptic equations with power type nonlinearities

We employ the enclosure method to reconstruct unknown inclusions within an object that is governed by a semilinear elliptic equation with power-type nonlinearity. Motivated by [16], we tried to solve the problem without using special solutions such as complex geometrical optics solutions to the equation. Instead, we construct approximate solutions obtained from Taylor approximation of the solution operator. By incorporating such approximate solutions into the definition of an indicator functional, we are able to use the classical Calderón-type harmonic functions to accomplish the reconstruction task.

math.AP↗

arXiv2017

Strong unique continuation for two-dimensional anisotropic elliptic systems

In this paper, we give the strong unique continuation property for a general two dimensional anisotropic elliptic system with real coefficients in a Gevrey class under the assumption that the principal symbol of the system has simple characteristics. The strong unique continuation property is derived by obtaining some Carleman estimate. The derivation of the Carleman estimate is based on transforming the system to a larger second order elliptic system with diagonal principal part which has complex coefficients.

math.AP↗

arXiv2015

The enclosure method for the anisotropic Maxwell system

We develop an enclosure-type reconstruction scheme to identify penetrable and impenetrable obstacles in electromagnetic field with anisotropic medium in \mathbb{R}^{3}. The main difficulty in treating this problem lies in the fact that there are so far no complex geometrical optics solutions available for the Maxwell's equation with anisotropic medium in \mathbb{R}^{3}. Instead, we derive and use another type of special solutions called oscillating-decaying solutions. To justify this scheme, we use Meyers' L^{p} estimate, for the Maxwell system, to compare the integrals coming from oscillating-decaying solutions and those from the reflected solutions.

math.AP↗

arXiv2011

Reconstruction of penetrable inclusions in elastic waves by boundary measurements

We use Ikehata's enclosure method to reconstruct penetrable unknown inclusions in a plane elastic body in time-harmonic waves. Complex geometrical optics solutions with complex polynomial phases are adopted as the probing utility. In a situation similar to ours, due to the presence of a zeroth order term in the equation, some technical assumptions need to be assumed in early researches. In a recent work of Sini and Yoshida, they succeeded in abandoning these assumptions by using a different idea to obtain a crucial estimate. In particular the boundaries of the inclusions need only to be Lipschitz. In this work we apply the same idea to our model. It's interesting that, with more careful treatment, we find the boundaries of the inclusions can in fact be assumed to be only continuous.

math.AP↗