arXiv subjects

Jianli Xiang

Publications and source records attributed to Jianli Xiang.

8 recordsLinked to original sources

arXiv2026

Limiting absorption principle for time-harmonic elastic scattering of plane waves from diffraction gratings

We establish the limiting absorption principle for time-harmonic elastic scattering of plane waves by a periodic rigid diffraction grating. By perturbing the frequency with a small positive imaginary part, we regularize the ill-posed problem at propagative wavenumbers (that is, when uniqueness fails under the classical Rayleigh expansion condition) and characterize the limiting solution via a singular perturbation result from functional analysis. The limiting solution satisfies the original scattering problem together with an additional constraint that ensures uniqueness. Both incident pressure and shear waves are considered, and the same constraint condition is obtained in both cases. The results provide a rigorous selection mechanism for physically admissible solutions at resonance frequencies. Our framework extends naturally to the Neumann (cavity) boundary condition as well as other transmission conditions, in particular when guided waves exist in periodic structures.

math.AP↗

arXiv2026

The monotonicity method for the inverse elastic scattering on unbounded domains

We discuss a time-harmonic inverse scattering problem for the Navier equation with compactly supported penetrable and possibly inhomogeneous scattering objects in an unbounded homogeneous background medium, and we develop a monotonicity relation for the far field operator that maps superpositions of incident plane waves to the far field patterns of the corresponding scattered waves. Combining the monotonicity relation with the method of localized potentials, we extend the so called monotonicity method to characterize the support of inhomogeneities in the Lamé parameters and the density in terms of the far field operator.

math.AP↗

arXiv2026

Monotonicity-based regularization of inverse medium scattering for shape reconstruction

We consider the scattering of time-harmonic plane waves by a compactly supported inhomogeneous scattering obstacle governed by the Helmholtz equation. Given far field observations of the scattered fields corresponding to plane wave incident fields for all possible incident and observation directions we study the inverse problem to recover the support of the scatterer. We propose a qualitative monotonicity-based regularization scheme which combines monotonicity-based shape reconstruction with one-step linearization to reconstruct a discrete approximation of the shape of the scatterer from noisy far field data. The purpose of the one-step linearization is to stabilize the monotonicity approach to shape reconstruction. We show that the monotonicity-based regularization scheme recovers the correct shape of the scatterer for noise-free data. Furthermore, we establish that the solution of the monotonicity-based regularization converges to the exact solution as the noise level tends to zero. We present numerical examples to illustrate our theoretical findings.

math.NA↗

arXiv2025

Absence of the analytic continuation of elastic transmission eigenfunctions at rectangular corners

We study time harmonic scattering problems in linear elasticity in $\mathbb{R}^{2}$. We show that certain penetrable scatterers with rectangular corners scatter every incident wave nontrivially. Even though these scatterers have interior transmission eigenvalues, the far field operator has a trivial kernel at every real frequency. Our approach relies on a special decomposition of the elastic Lamé operator and also provides an alternative idea for treating inverse elastic medium problems with a general polygonal support.

math.AP↗

arXiv2025

Uniqueness in determining a convex polygonal source of an elastic body

In this work, we consider the time-harmonic inverse elastic source problem of a fixed frequency for the Navier equation in two dimensions. We show that a convex polygon can be uniquely determined by a single far field measurement. Our approach relies on the corner singularity analysis of solutions to the inhomogeneous Navier equation with a source term in a sector. This paper also contributes to corner scattering theory for the Navier equation in an non-convex domain.

math.AP↗

arXiv2024

Uniqueness to inverse acoustic and elastic medium scattering problems with hyper-singular source method

This paper is concerned with inverse scattering problems of determining the support of an isotropic and homogeneous penetrable body from knowledge of multi-static far-field patterns in acoustics and in linear elasticity. The normal derivative of the total fields admits no jump on the interface of the scatterer in the trace sense. If the contrast function of the refractive index function or the density function has a positive lower bound near the boundary, we propose a hyper-singular source method to prove uniqueness of inverse scattering with all incoming plane waves at a fixed energy. It is based on subtle analysis on the leading part of the scattered field when hyper-singular sources caused by the first derivative of the fundamental solution approach to a boundary point. As a by-product, we show that this hyper-singular method can be also used to determine the boundary value of a Holder continuous refractive index function in acoustics or a Holder continuous density function in linear elasticity.

math.AP↗

arXiv2023

Uniqueness in determining rectangular grating profiles with a single incoming wave (Part II): TM polarization case

This paper is concerned with an inverse transmission problem for recovering the shape of a penetrable rectangular grating sitting on a perfectly conducting plate. We consider a general transmission problem with the coefficient λ\neq 1 which covers the TM polarization case. It is proved that a rectangular grating profile can be uniquely determined by the near-field observation data incited by a single plane wave and measured on a line segment above the grating. In comparision with the TE case (λ=1), the wave field cannot lie in H^2 around each corner point, bringing essential difficulties in proving uniqueness with one plane wave. Our approach relies on singularity analysis for Helmholtz transmission problems in a right-corner domain and also provides an alternative idea for treating the TE transmission conditions which were considered in the authors' previous work [Inverse Problem, 39 (2023): 055004.]

math.AP↗

arXiv2021

Uniqueness in determining binary grating profiles and refractive indices with a single incoming wave

We investigate inverse diffraction problems for penetrable gratings in a piecewise constant medium. In the TE polarization case, it is proved that a binary grating profile together with the refractive index beneath it can be uniquely determined by the near-field observation data incited by a single plane wave and measured on a line segment above the grating. Our approach relies on the expansion of solutions to the Helmholtz equation and the corner singularity analysis of solutions to the inhomogeneous Laplace equation with a piecewise continuous source term in a sector. This paper also contributes to corner scattering theory for the Helmholtz equation in a special non-convex domain.

math.AP↗