Finite element methods for Maxwell's equations.

In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

A recently developed formulation of the inverse source problem as a Fredholm integral equation of the first kind provides motivation for the development of analytical characterizations of the nonuniqueness in the inverse source problem. Nonradiating sources, i. e., sources for which the field is identically zero outside a finite region, are introduced. It is then shown that the null space of the Fredholm integral equation is exactly the class of nonradiating sources.

Non-uniqueness in the inverse source problem in acoustics and electromagnetics

Uniqueness and reconstruction for the fractional Calderón problem with a single measurement

The Calderon problem for the fractional Schrodinger equation was introduced in the work Ghosh et al. (to appear) which gave a global uniqueness result also in the partial data case. This article improves this result in two ways. First, we prove a quantitative uniqueness result showing that this inverse problem enjoys logarithmic stability under suitable a priori bounds. Second, we show that the results are valid for potentials in scale-invariant L p or negative order Sobolev spaces. A key point is a quantitative approximation property for solutions of fractional equations, obtained by combining a careful propagation of smallness analysis for the Caffarelli–Silvestre extension and a duality argument.

The fractional Calderón problem: Low regularity and stability

In this paper, we introduce an inverse problem of a Schrodinger type variable nonlocal elliptic operator (−∇⋅(A(x)∇))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≥2. Our results generalize the recent initiative [18] of introducing and solving inverse problem for fractional Schrodinger 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.

The Calderón problem for variable coefficients nonlocal elliptic operators

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

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.

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

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

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 $\mathbf{F}_j\in C^\alpha(\overline{\mathcal{K}^{r_0}_{x_0}}; \mathbb{C}^3)$, $j=1,2$, where $\alpha$ is the positive Holder index. Consider the following electromagnetic problem $$\left\{\begin{split} & \nabla\wedge \mathbf{E}-\mathrm{i}\omega \mu_0 \mathbf{H}=\mathbf{F}_{1} \quad \mbox{in $\mathcal{K}^{r_0}_{x_0}$},\\ & \, \nabla\wedge \mathbf{H}+\mathrm{i}\omega\varepsilon_0 \mathbf{E}=\mathbf{F}_{2} \quad \mbox{in $\mathcal{K}^{r_0}_{x_0}$}, \\ &\, \nu\wedge \mathbf{E}=\nu\wedge\mathbf{H}=0 \qquad\mbox{on $\partial \mathcal{K}^{r_0}_{x_0}\setminus \partial B_{r_0}(x_0)$}, \end{split}\right.$$ where $\nu$ 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.

Jingni Xiao

Papers

The Calderón problem for variable coefficients nonlocal elliptic operators

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

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

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

On an electromagnetic problem in a corner and its applications