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Inverse Boundary Problems for Elliptic Operators in Low Regularity Setting: Uniqueness and Stability Issues


In this dissertation we study inverse boundary problems for elliptic PDE in the low regularity setting. The new results established are concerned with the uniqueness and stability issues. Specifically, in Chapter 2 we analyze inverse boundary problems for first order perturbations of the Laplacian, which arise as model operators in the acoustic tomography of a moving fluid. We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded domain in \mathbb{R}^n, n \geq 3, determines the first order perturbation of low regularity up to a natural gauge transformation, which sometimes is trivial. As an application, we recover the fluid parameters of low regularity from boundary measurements, allowing some of them to be discontinuous.

In Chapter 3, we study inverse boundary problems of determining a potential in the Helmholtz type equation for the perturbed biharmonic operator from the knowledge of the partial Cauchy data set. Our geometric setting is that of a domain whose inaccessible portion of the boundary is contained in a hyperplane, and we are given the Cauchy data set on the complement. We establish stability estimates in the high frequency regime, with an explicit dependence on the frequency parameter, under mild regularity assumptions on the potentials.

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