"The last fifteen years have seen much success in the analysis of quasi-static evolution for Griffith fracture, which is the mathematically natural starting point for studying fracture. At the same time, attempts have been made to show existence for similar models based on cohesive fracture rather than Griffith. These models are generally viewed as physically more realistic than Griffith, in that they are better models for crack nucleation. These attempts at existence proofs have been unsuccessful without very strong additional assumptions, for example, specifying the crack path a priori. The main purpose of this thesis is to characterize as well as possible the mathematical difficulties in cohesive fracture, and to make progress toward an existence result without the prescribed crack path assumption. So far, the most powerful method for existence proofs is to build a sequence of approximate solutions, based on time discretization, and take the limit as the time steps go to zero. We show that there are mainly two complications on the cracks of these approximate solutions that we need to rule out in order to show existence. The first one is due to the potential oscillation of the crack path. The second is due to the potential splitting of a crack into two or more nearby cracks, with the same total jump in displacement. We begin by first constructing an example illustrating how oscillations described above can affect the minimality of the limit. Then we prove that the splitting described above can be ruled out for any sequence of unilateral minimizers. With this result, we show how exactly oscillation affect the minimality on the limit of the sequence. We then move to the evolution problem and show the convergence of energy for almost every t. Based on this result we develop a method that allows us to analyze the problem using only a finite set of times. An application of this method is a proof of absolute continuity. Future work will be aimed at using the tools we developed to rule out oscillation and finally to prove existence results under more general assumptions."

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.

4 Summary Altogether, the six most common bee viruses which infect the honey bee (Apis mellifera) were monitored in the territory of the Czech Republic between 2006 and 2009. Parallel infections of viruses (DWV, ABPV and BQCV) in bee adults and parallel co- infection of viruses with fungal diseases caused by Nosema apis and Nosema ceranae were confirmed by PCR tests. A new sensitive method of detection of the originator of the American foulbrood (Paenibacillus larvae) from bee debris was developed for the practical use of detection of AFB disease in bee populations. Various approaches for the extraction of spores from bee debris and lyses of spores were compared. The sensitivity of PCR tests for the presence of Paenibacillus larvae in debris was compared with the classic cultivation method. The PCR method for the detection American foulbrood was further studied and developed to be more efficient. A new method, based on a matrix-like sample re-arrangement and a use of pooled samples, has been developed for testing 1000 samples in 35 PCR reactions. Another goal was to develop a robust and fast screening method for American foulbrood based on the cultivation test using paper sheets RIDA®COUNT with a specific cultivation medium, specific selection conditions for Paenibacillus larvae and chromogen visualization...