Finite element methods for Maxwell's equations.

Preface 1. Fourier series and Fourier transforms 2. Distributions 3. Elliptic equations (fundamental theory) 4. Initial value problems (Cauchy problems) 5. Evolution equations 6. Hyperbolic equations 7. Semi-linear hyperbolic equations 8. Green's functions and spectra Supplementary remarks Guide to the literature Bibliography Symbols Index.

The theory of partial differential equations

Proceedings of the American Mathematical Society

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Integral Equation Methods In Scattering Theory

This chapter is concerned with the introduction of the basic integral equations for two-dimensional elastic linear material problems. It starts by briefly reviewing the partial differential equations for linear elastic material and introducing the necessary notations involved in the formulation. These governing equations are also extended to deal with problems in which initial stress and strain type loads are applied. Such kind of loads are not only important to take into account temperature or other similar loads, but also to model nonlinear material behaviour when used in conjunction with a well established successive elastic solution technique.

Boundary Integral Equations

Let \begin{document}$A∈{\rm{Sym}}(n× n)$\end{document} be an elliptic 2-tensor. Consider the anisotropic fractional Schrodinger operator \begin{document}$\mathscr{L}_A^s+q$\end{document} , where \begin{document}$\mathscr{L}_A^s: = (-\nabla·(A(x)\nabla))^s$\end{document} , \begin{document}$s∈ (0, 1)$\end{document} and \begin{document}$q∈ L^∞$\end{document} . We are concerned with the simultaneous recovery of \begin{document}$q$\end{document} and possibly embedded soft or hard obstacles inside \begin{document}$q$\end{document} by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain \begin{document}$Ω$\end{document} associated with \begin{document}$\mathscr{L}_A^s+q$\end{document} . It is shown that a single measurement can uniquely determine the embedded obstacle, independent of the surrounding potential \begin{document}$q$\end{document} . If multiple measurements are allowed, then the surrounding potential \begin{document}$q$\end{document} can also be uniquely recovered. These are surprising findings since in the local case, namely \begin{document}$s = 1$\end{document} , both the obstacle recovery by a single measurement and the simultaneous recovery of the surrounding potential by multiple measurements are long-standing problems and still remain open in the literature. Our argument for the nonlocal inverse problem is mainly based on the strong uniqueness property and Runge approximation property for anisotropic fractional Schrodinger operators.

Simultaneously recovering potentials and embedded obstacles for anisotropic fractional schrödinger operators

We study various partial data inverse boundary value problems for the semilinear elliptic equation $\Delta u+ a(x,u)=0$ in a domain in $\mathbb R^n$ by using the higher order linearization technique introduced in [LLS 19, FO19]. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of $a(x,z)$ at $z=0$ under general assumptions on $a(x,z)$. The determination of the Taylor series can be done in parallel with the detection of an unknown cavity inside the domain or an unknown part of the boundary of the domain. The method relies on the solution of the linearized partial data Calderon problem [FKSU09], and implies the solution of partial data problems for certain semilinear equations $\Delta u+ a(x,u) = 0$ also proved in [KU19]. 
The results that we prove are in contrast to the analogous inverse problems for the linear Schrodinger equation. There recovering an unknown cavity (or part of the boundary) and the potential simultaneously are long-standing open problems, and the solution to the Calderon problem with partial data is known only in special cases when $n \geq 3$.

Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations

We consider the inverse problems of for the fractional Schr\"{o}dinger equation by using monotonicity formulas. We provide if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal Dirichlet-to-Neumann maps. Based on the monotonicity relation, we can prove uniqueness for the nonlocal Calder\'{o}n problem in a constructive manner. Secondly, we offer a reconstruction method for an unknown obstacles in a given domain. Our method is independent of the dimension $n\geq 2$ and only requires the background solution of the fractional Schr\"{o}dinger equation.

Monotonicity-based inversion of the fractional Schr\"odinger equation

This paper is concerned with the forward and inverse problems for the fractional semilinear elliptic equation $(-\Delta)^s u +a(x,u)=0$ for $0<s<1$. For the forward problem, we proved the problem is well-posed and has a unique solution for small exterior data. The inverse problems we consider here consists of two cases. First we demonstrate that an unknown coefficient $a(x,u)$ can be uniquely determined from the knowledge of exterior measurements, known as the Dirichlet-to-Neumann map. Second, despite the presence of an unknown obstacle in the media, we show that the obstacle and the coefficient can be recovered concurrently from these measurements. Finally, we investigate that these two fractional inverse problems can also be solved by using a single measurement, and all results hold for any dimension $n\geq 1$.

Inverse problems for fractional semilinear elliptic equations

In this work, we use monotonicity-based methods for the fractional Schrodinger equation with general potentials q in L^\infty(Omega) in a Lipschitz bounded open set Omega \subset R^n in any dimensi...

Yi-Hsuan Lin

Papers

Simultaneously recovering potentials and embedded obstacles for anisotropic fractional schrödinger operators

Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations

Monotonicity-based inversion of the fractional Schr\"odinger equation

Inverse problems for fractional semilinear elliptic equations

Monotonicity-based inversion of the fractional schodinger equation ii. general potentials and stability