Variations autour de formes irrégulières et optimales

Abstract : This dissertation takes place in the mathematic field called shape optimization. More precisely, we focus on difficulties linked to the writing of optimality conditions, and how to use them. The two main obstacles that have been analysed are the following:
- to deal with shape whose regularity is a priori unknown,
- to deal with strong geometrical constraints, i.e. which allow very few variations in the writing of optimality (for example the convexity).

The results are described in the four chapters of the thesis:
- the first one aims at developing a framework of shape derivatives, well adapted for shapes with very poor regularity,
- the chapter 2 deals with the analysis of optimality conditions under convexity constraints, in dimension 2, and their applications to a class of problems whose solutions are necessarily polygons,
- the third one focuses on two classical problems of shape optimization for eigenvalues, which enlighten the difficulties previously mentioned. We prove some regularity results, and also non-regularity ones, of optimal shapes for these problems; we get some maximal regularity in $\C^{1,1/2}$, which are new and sharp,
- the last chapter is motivated by the question of partially overdetermined problems, and we build some counter-examples linked with shape optimization.
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Submitted on : Thursday, December 11, 2008 - 1:39:33 PM
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Jimmy Lamboley. Variations autour de formes irrégulières et optimales. Mathématiques [math]. École normale supérieure de Cachan - ENS Cachan, 2008. Français. ⟨tel-00346316⟩

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