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

Inverse problems for elliptic equations with power type nonlinearities

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

We investigate the Calderon problem for the fractional Schrodinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does not enjoy a gauge invariance. The uniqueness result is complemented by an associated logarithmic stability estimate under suitable apriori assumptions. Also uniqueness under finitely many generic measurements is discussed. Here the genericity is obtained through singularity theory which might also be interesting in the context of hybrid inverse problems. Combined with the results from Ghosh et al. (Uniqueness and reconstruction for the fractional Calderon problem with a single easurement, 2018. arXiv:1801.04449), this yields a finite measurements constructive reconstruction algorithm for the fractional Calderon problem with drift. The inverse problem is formulated as a partial data type nonlocal problem and it is considered in any dimension $$n\ge 1$$.

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The Calderón problem for the fractional Schrödinger equation with drift

We study various partial data inverse boundary value problems for the semilinear elliptic equation Δu+a(x,u)=0 in a domain in Rn by using the higher order linearization technique introduced by Lassas–Liimatainen–Lin–Salo and Feizmohammadi–Oksanen. 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 by Ferreira–Kenig–Sjostrand–Uhlmann, and implies the solution of partial data problems for certain semilinear equations Δu+a(x,u)=0 also proved by Krupchyk–Uhlmann.

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

We study global uniqueness in an inverse problem for the fractional semilinear Schrödinger equation (−∆)su + q(x, u) = 0 with s ∈ (0, 1). We show that an unknown function q(x, u) can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to 2. Moreover, we demonstrate the comparison principle and provide a L∞ estimate for this nonlocal equation under appropriate regularity assumptions.

Yi-Hsuan Lin

Papers

Inverse problems for elliptic equations with power type nonlinearities

The Calderón problem for variable coefficients nonlocal elliptic operators

The Calderón problem for the fractional Schrödinger equation with drift

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

Global uniqueness for the fractional semilinear Schrödinger equation