This thesis is divided into two parts. The first is devoted to the study of image restorationand the second part is devoted to the study of a Frenkel-Kontorova model using methodsfrom the calculus of variations and partial differential equations. In chapter 1, we presentthe key issues we will discuss in this thesis, and recal the denitions and some properties ofspaces of functions of bounded variations BV , Orlicz Sobolev spaces and Frenkel-Kontorovamodel results on image analysis. In chapter 2, we show that the non-convex minimizationproblems (restoration image) involving sublinear regularizing terms are ill-posed. In chapter3, we study a model of restoration with nonstandard increasing regularizing terms,proposedby Blomgren. We show that is lower semi-continuous in the weak topologie of some Sobolev-Orlicz space associated with it, which allows existence result of the solution. In Chaptre 4, we study a Frenkel-Kontorova model, that we show existence of at least a traveling wave type solution, u.
