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On some applications of functions of bounded variation in finite and infinite dimension

Abstract : The aim of this thesis is to investigate some applications of the functions of bounded variation and sets of finite perimeter. We mainly focus on applications in image processing, geometry and infinite dimensional analysis. We study first a Primal-Dual method proposed by Appleton and Talbot for solving some imaging problems. We give a new interpretation of this method which leads to a better mathematical understanding. This enables us for example to prove the convergence of the method and give new a posteriori estimates which are very important for numerical use. We then consider the problem of prescribed mean curvature surfaces in periodic environment. Using the theory of sets of finite perimeter, we prove existence of compact approximated solutions to this problem. We also study the asymptotic behavior of these solutions when their volume goes to infinity. The last two parts of the thesis are devoted to the study of some geometric problems in Wiener spaces. Studying on the one hand, the relationship between symmetrization, semi-continuity and isoperimetric inequalities, we compute the relaxation of the perimeter in this infinite dimensional setting and give an elliptic approximation of this lower semicontinuous envelope. On the other hand, we show convexity of the minimizers for some variational problems in Wiener spaces. One of the main ingredients in this study is the generalization of representations formulas for integral functionals in this setting.
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Submitted on : Thursday, January 19, 2012 - 2:23:04 PM
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Michael Goldman. On some applications of functions of bounded variation in finite and infinite dimension. Analysis of PDEs [math.AP]. Ecole Polytechnique X, 2011. English. ⟨NNT : 2011EPXX0058⟩. ⟨pastel-00661393⟩

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