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Jingni Xiao

Publications and source records attributed to Jingni Xiao.

14 recordsLinked to original sources
arXiv2026

Split Corners in Scattering and Inverse Scattering

We study scattering and inverse scattering generated by sources and penetrable media with corner singularities. We introduce the notion of split corners, a local model that unifies geometric singularities and coefficient discontinuities arising in source and medium scattering. Within this framework, we establish scattering and inverse scattering results that extend the classical theory to split corners, allowing the source or medium contrast to approach different limiting values in distinct sectors meeting at the corner tip. For source scattering, we prove that every admissible two-split corner necessarily radiates in a general bounded inhomogeneous background, thereby generalizing the classical corner radiation principle to configurations involving both geometric corners and jump discontinuities. For penetrable media, we establish analogous results together with explicit compatibility conditions for incident waves of arbitrary vanishing order. As consequences, we obtain several uniqueness results in inverse source and inverse medium scattering, including recovery of polygonal convex hulls from a single far-field measurement.

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arXiv2026

Finiteness of Nonscattering Wavenumbers for Herglotz Incident Waves

This paper continues the study initiated in \cite{VogXia25} on nonscattering phenomena for inhomogeneous media. We investigate star-shaped domains in $\mathbb{R}^2$ and establish finiteness results for nonscattering wavenumbers associated with Herglotz incident waves of fixed density. First, for ellipses we establish finiteness for all constant contrasts $q\neq 1$, removing the geometric restrictions required in previous work. Second, for admissible star-shaped domains with $q\in(0,1)$, we introduce a flexible interval-wise geometric framework that unifies and generalizes earlier finiteness results. Our results reveal that infinite sequences of nonscattering wavenumbers are tied to exact radial symmetry and cannot persist under admissible geometric perturbations. %extend those of \cite{VogXia25} to the regime $0<q<1$ and reveal a rigidity phenomenon for nonscattering behavior beyond the radially symmetric setting.

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arXiv2024

Examples of non-scattering inhomogeneities

We consider the scattering of waves by a penetrable inclusion embedded in some reference medium. We exhibit examples of materials and geometries for which non-scattering frequencies exist, i.e., for which at some frequencies there are incident fields which produce null scattered fields outside of the inhomogeneity. We show in particular that certain domains with corners or even cusps can support non-scattering frequencies. We relate the latter, for some inclusions, to resonance frequencies for Dirichlet or Neumann cavities. We also find situations where incident non-scattering fields solve the Helmholtz equation in a neighborhood of the inhomogeneity and not in the whole space. Finally, in relation with invisibility, we give examples of inclusions of anisotropic materials which are non-scattering for all real frequencies. We prove that corresponding material indices must have a special structure on the boundary.

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arXiv2022

On the Regularity of Non-Scattering Anisotropic Inhomogeneities

In this paper we examine necessary conditions for an anisotropic inhomogeneous medium to be non-scattering at a single wave number and for a single incident field. These conditions are expressed in terms of the regularity of the boundary of the inhomogeneity. We assume that the coefficients, characterizing the constitutive material properties of the medium, are sufficiently smooth, and the incident wave is appropriately non-degenerate. Our analysis utilizes the Hodograph transform as well as regularity results for nonlinear elliptic partial differential equations. Our approach requires that the boundary a-priori is of class $C^{1,α}$ for some $0<α<1$.

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arXiv2021

A New Type of CGO Solutions and its Applications in Corner Scattering

We consider corner scattering for the operator $\nabla \cdot γ(x)\nabla +k^2ρ(x)$ in $\mathbb{R}^2$, with $γ$ a positive definite symmetric matrix and $ρ$ a positive scalar function. A corner is referred to one that is on the boundary of the (compact) support of $γ(x)-I$ or $ρ(x)-1$, where $I$ stands for the identity matrix. We assume that $γ$ is a scalar function in a small neighborhood of the corner. We show that any admissible incident field will be scattered by such corners, which are allowed to be concave. Moreover, we provide a brief discussion on the existence of non-scattering waves when $γ-I$ has a jump across the corner. In order to prove the results, we construct a new type of complex geometric optics (CGO) solutions.

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arXiv2020

On Corner Scattering for Operators of Divergence Form and Applications to Inverse Scattering

We consider the scattering problem governed by the Helmholtz equation with inhomogeneity in both `conductivity' in the divergence form and `potential' in the lower order term. The support of the inhomogeneity is assumed to contain a convex corner. We prove that, due to the presence of such corner under appropriate assumptions on the potential and conductivity in the vicinity of the corner, any incident field scatters. Based on corner scattering analysis we present a uniqueness result on determination of the polygonal convex hull of the support of admissible inhomogeneities, from scattering data corresponding to one single incident wave. These results require only certain regularity around the corner for the coefficients modeling the inhomogeneity, whereas away from the corner they can be quite general. Our main results on scattering and inverse scattering are established for $\mathbb{R}^2$, while some analytic tools are developed in any dimension $n\geq 2$.

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arXiv2019

On an electromagnetic problem in a corner and its applications

Let $\mathcal{K}^{r_0}_{x_0}$ be a (non-degenerate) truncated corner in $\mathbb{R}^3$ with $x_0\in\mathbb{R}^3$ being its apex, and $\mathbf{F}_j\in C^α(\overline{\mathcal{K}^{r_0}_{x_0}}; \mathbb{C}^3)$, $j=1,2$, where $α$ is the positive Hölder index. Consider the following electromagnetic problem $$\left\{\begin{split} & \nabla\wedge \mathbf{E}-\mathrm{i}ωμ_0 \mathbf{H}=\mathbf{F}_{1} \quad \mbox{in $\mathcal{K}^{r_0}_{x_0}$},\\ & \, \nabla\wedge \mathbf{H}+\mathrm{i}ω\varepsilon_0 \mathbf{E}=\mathbf{F}_{2} \quad \mbox{in $\mathcal{K}^{r_0}_{x_0}$}, \\ &\, ν\wedge \mathbf{E}=ν\wedge\mathbf{H}=0 \qquad\mbox{on $\partial \mathcal{K}^{r_0}_{x_0}\setminus \partial B_{r_0}(x_0)$}, \end{split}\right.$$ where $ν$ denotes the exterior unit normal vector of $\partial \mathcal{K}^{r_0}_{x_0}$. We prove that $\mathbf{F}_1$ and $\mathbf{F}_2$ must vanish at the apex $x_0$. There are a series of interesting consequences of this vanishing property in several separate but intriguingly connected topics in electromagnetism. First, we can geometrically characterize non-radiating sources in time-harmonic electromagnetic scattering. Secondly, we consider the inverse source scattering problem for time-harmonic electromagnetic waves and establish the uniqueness result in determining the polyhedral support of a source by a single far-field measurement. Thirdly, we derive a property of the geometric structure of electromagnetic interior transmission eigenfunctions near corners. Finally, we also discuss its implication to invisibility cloaking.

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arXiv2018

Revisiting the Decoupling of Elastic Waves From a Weak Formulation Perspective

Elastic scattering governed by the Lame system associated with the third-type or fourth-type boundary condition is considered. It was shown in [8] by two of the authors that under suitable geometric conditions on the boundary surface of the elastic inclusion, the longitudinal and shear waves can be decoupled. The decoupling result in [8] was derived based on analyzing the local boundary behaviours of the elastic fields. In this article, we provide a different argument from a variational perspective in proving the decoupling result.

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arXiv2017

Mosco convergence for H(curl) spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems

This paper is concerned with the scattering problem for time-harmonic electromagnetic waves, due to the presence of scatterers and of inhomogeneities in the medium. We prove a sharp stability result for the solutions to the direct electromagnetic scattering problem, with respect to variations of the scatterer and of the inhomogeneity, under minimal regularity assumptions for both of them. The stability result leads to uniform bounds on solutions to the scattering problems for an extremely general class of admissible scatterers and inhomogeneities. The uniform bounds are a key step to tackle the challenging stability issue for the corresponding inverse electromagnetic scattering problem. In this paper we establish two optimal stability results of logarithmic type for the determination of polyhedral scatterers by a minimal number of electromagnetic scattering measurements. In order to prove the stability result for the direct electromagnetic scattering problem, we study two fundamental issues in the theory of Maxwell equations: Mosco convergence for H(curl) spaces and higher integrability properties of solutions to Maxwell equations in nonsmooth domains.

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arXiv2017

The Calderón problem for variable coefficients nonlocal elliptic operators

In this paper, we introduce an inverse problem of a Schrödinger type variable nonlocal elliptic operator $(-\nabla\cdot(A(x)\nabla))^{s}+q)$, for $0<s<1$. We determine the unknown bounded potential $q$ from the exterior partial measurements associated with the nonlocal Dirichlet-to-Neumann map for any dimension $n\geq2$. Our results generalize the recent initiative [16] of introducing and solving inverse problem for fractional Schrödinger operator $((-Δ)^{s}+q)$ for $0<s<1$. We also prove some regularity results of the direct problem corresponding to the variable coefficients fractional differential operator and the associated degenerate elliptic operator.

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arXiv2016

On electromagnetic scattering from a penetrable corner

This article is concerned with the time-harmonic electromagnetic (EM) scattering from a generic inhomogeneous medium. It is shown that if there is a right corner on the support of the medium, then it scatters every pair of incident EM fields, excluding a possible class of EM fields which are of very particular forms. That is, for every pair of admissible incident EM fields, the corresponding scattered wave fields associated to the medium scatterer cannot be identically vanishing outside the support of the medium. Indeed, we achieve the corner scattering result by establishing a stronger result, that shows the failure of the analytic extension across the corner of certain EM fields satisfying the so-called interior transmission eigenvalue problem. This extends the relevant study in [3] for the acoustic scattering governed by the Helmholtz equation to the electromagnetic case governed by the Maxwell system. Substantial new challenges arise from the corresponding extension from the scalar PDE to the system of PDEs. Our mathematical arguments combine the analysis for interior transmission eigenvalue problems associated to the Maxwell system; the derivation of novel orthogonality relation for the solutions of Maxwell systems; the construction of complex-geometrical-optics (CGO) solutions for the Maxwell system with new Lp-estimates (p > 6) on the remainder terms; and the proof of the non-vanishing property for the Laplace transform of vectorial homogeneous harmonic polynomials.

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arXiv2016

Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements

The aim of the paper is to establish optimal stability estimates for the determination of sound-hard polyhedral scatterers in $\mathbb{R}^N$, $N \geq 2$, by a minimal number of far-field measurements. This work is a significant and highly nontrivial extension of the stability estimates for the determination of sound-soft polyhedral scatterers by far-field measurements, proved by one of the authors, to the much more challenging sound-hard case. The admissible polyhedral scatterers satisfy minimal a priori assumptions of Lipschitz type and may include at the same time solid obstacles and screen-type components. In this case we obtain a stability estimate with $N$ far-field measurements. Important features of such an estimate are that we have an explicit dependence on the parameter $h$ representing the minimal size of the cells forming the boundaries of the admissible polyhedral scatterers, and that the modulus of continuity, provided the error is small enough with respect to $h$, does not depend on $h$. If we restrict to $N=2,3$ and to polyhedral obstacles, that is to polyhedra, then we obtain stability estimates with fewer measurements, namely first with $N-1$ measurements and then with a single measurement. In this case the dependence on $h$ is not explicit anymore and the modulus of continuity depends on $h$ as well.

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arXiv2015

Decoupling Elastic Waves and Its Applications

In this paper, we consider time-harmonic elastic wave scattering governed by the Lamé system. It is known that the elastic wave field can be decomposed into the shear and compressional parts, namely, the pressure and shear waves that are generally coexisting, but propagating at different speeds. We consider the third or fourth kind scatterer and derive two geometric conditions, respectively, related to the mean and Gaussian curvatures of the boundary surface of the scatterer that can ensure the decoupling of the shear and pressure waves. Then we apply the decoupling results to the uniqueness and stability analysis for inverse elastic scattering problems in determining polyhedral scatterers by a minimal number of far-field measurements.

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