This thesis consists of six scientific papers, an introduction and a summary. All six papers concern the boundary behavior of non-negative solutions to partial differential equations. Paper I concerns solutions to certain p-Laplace type operators with variable coefficients. Suppose that u is a non-negative solution that vanishes on a part Γ of an Ahlfors regular NTA-domain. We prove among other things that the gradient Du of u has non-tangential limits almost everywhere on the boundary piece Γ, and that log|Du| is a BMO function on the boundary.  Furthermore, for Ahlfors regular NTA-domains that are uniformly (N,δ,r0)-approximable by Lipschitz graph domains we prove a boundary Harnack inequality provided that δ is small enough.  Paper II concerns solutions to a p-Laplace type operator with lower order terms in δ-Reifenberg flat domains. We prove that the ratio of two non-negative solutions vanishing on a part of the boundary is Hölder continuous provided that δ is small enough. Furthermore we solve the Martin boundary problem provided δ is small enough. In Paper III we prove that the boundary type Riesz measure associated to an A-capacitary function in a Reifenberg flat domain with vanishing constant is asymptotically optimal doubling. Paper IV concerns the boundary behavior of solutions to certain parabolic equations of p-Laplace type in Lipschitz cylinders. Among other things, we prove an intrinsic Carleson type estimate for the degenerate case and a weak intrinsic Carleson type estimate in the singular supercritical case. In Paper V we are concerned with equations of p-Laplace type structured on Hörmander vector fields. We prove that the boundary type Riesz measure associated to a non-negative solution that vanishes on a part Γ of an X-NTA-domain, is doubling on Γ. Paper VI concerns a one-phase free boundary problem for linear elliptic equations of non-divergence type. Assume that we know that the positivity set is an NTA-domain and that the free boundary is a graph. Furthermore assume that our solution is monotone in the graph direction and that the coefficients of the equation are constant in the graph direction. We prove that the graph giving the free boundary is Lipschitz continuous.

Cette thèse est une contribution au problème de détection de fines structures tubulaires dans une image2-D ou 3-D. Nous avons plus précisément en vue le cas des images angiographiques. Celles-ci étant bruitées, les vaisseaux ne se détachent pas nettement du reste de l’image, la question est donc de segmenter avec précision le réseau sanguin. Le cadre théorique de ce travail est le calcul des variations eten particulier l'énergie de Mumford-Shah. Cependant, ce modèle n'est adapté qu'à la détection de structures volumiques étendues dans toutes les directions de l’image. Le but de ce travail est donc deconstruire une énergie qui favorise les ensembles qui ne sont étendus que dans une seule direction, cequi est le cas de fins tubes. Pour cela, une nouvelle inconnue est introduite, une métrique Riemannienne,qui a pour but la détection de la structure géométrique de l’image. Une nouvelle formulation de l’énergie de Mumford-Shah est donnée avec cette nouvelle métrique. La preuve de l'existence d'une solution au problème de la minimisation de l’énergie est apportée. De plus, une approximation par gamma-convergence est démontrée, ce qui permet ensuite de proposer et de mettre en oeuvre une implémentation numérique. / This thesis is a contribution to the fine tubular structures detection problem in a 2-D or 3-D image. We arespecifically interested in the case of angiographic images. The vessels are surrounded by noise and thenthe question is to segment precisely the blood network. The theoretical framework of our work is thecalculus of variations and we focus on the Mumford-Shah energy. Initially, this model is adapted to thedetection of volumetric structures extended in all directions of the image. The aim of this study is to buildan energy that favors sets which are extended in one direction, which is the case of fine tubes. Then, weintroduce a new unknown, a Riemannian metric, which captures the geometric structure at each point ofthe image and we give a new formulation of the Mumford-Shah energy adapted to this metric. Thecomplete analysis of this model is done: we prove that the associated problem of minimization is wellposed and we introduce an approximation by gamma-convergence more suitable for numerics. Eventually,we propose numerical experimentations adapted to this approximation.