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Title: On local constraints and regularity of PDE in electromagnetics : applications to hybrid imaging inverse problems
Author: Alberti, Giovanni S.
ISNI:       0000 0004 5354 5944
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2014
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The first contribution of this thesis is a new regularity theorem for time harmonic Maxwell's equations with less than Lipschitz complex anisotropic coefficients. By using the Lp theory for elliptic equations, it is possible to prove H1 and Hölder regularity results, provided that the coefficients are W1,p for some p = 3. This improves previous regularity results, where the assumption W1,∞ for the coefficients was believed to be optimal. The method can be easily extended to the case of bi-anisotropic materials, for which a separate approach turns out to be unnecessary. The second focus of this work is the boundary control of the Helmholtz and Maxwell equations to enforce local constraints inside the domain. More precisely, we look for suitable boundary conditions such that the corresponding solutions and their derivatives satisfy certain local non-zero constraints. Complex geometric optics solutions can be used to construct such illuminations, but are impractical for several reasons. We propose a constructive approach to this problem based on the use of multiple frequencies. The suitable boundary conditions are explicitly constructed and give the desired constraints, provided that a finite number of frequencies, given a priori, are chosen in a fixed range. This method is based on the holomorphicity of the solutions with respect to the frequency and on the regularity theory for the PDE under consideration. This theory finds applications to several hybrid imaging inverse problems, where the unknown coefficients have to be imaged from internal measurements. In order to perform the reconstruction, we often need to find suitable boundary conditions such that the corresponding solutions satisfy certain non-zero constraints, depending on the particular problem under consideration. The multiple frequency approach introduced in this thesis represents a valid alternative to the use of complex geometric optics solutions to construct such boundary conditions. Several examples are discussed.
Supervisor: Capdeboscq, Yves Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Functional analysis (mathematics) ; Numerical analysis ; Optics,electromagnetic theory (mathematics) ; Partial differential equations ; Regularity theory ; Maxwell's equations ; inverse problems ; hybrid imaging ; medical imaging