Shape optimization problems of higher codimension

The field of shape optimization problems has received a lot of attention in recent years, particularly in relation to a number of applications in physics and engineering that require a focus on shapes instead of parameters or functions. In general for ap- plications the aim is to deform and modify the admissible shapes in order to optimize a given cost function. The fascinating feature is that the variables are shapes, i.e., domains of R^{d}, instead of functions. This choice often produces additional dicul- ties for the existence of a classical solution (that is an optimizing domain) and the introduction of suitable relaxed formulation of the problem is needed in order to get a solution which is in this case a measure. However, we may obtain a classical solution by imposing some geometrical constraint on the class of competing domains or requiring the cost functional verifies some particular conditions. The shape optimization problem is in general an optimization problem of the form min\{F(\Omega): \Omega \in {\cal O} \}; where F is a given cost functional and {\cal O} a class of domains in R^{d}. They are many books written on shape optimization problems. The thesis is organized as follows: the first chapter is dedicated to a brief introduction and presentation of some examples. In Academic examples, we present the isoperimetric problems, minimal and capillary surface problems and the spectral optimization problems while in applied examples the Newton's problem of optimal aerodinamical profile and optimal mixture of two con- ductors are considered. The second chapter is concerned with some basics elements of geometric measure theory that will be used in the sequel. After recalling some notions of abstract measure theory, we deal with the Hausdorff measures which are important for defining the notion of approximate tangent space. Finally we introduce the notion of approximate tangent space to a measure and to a set and also some differential op- erators like tangential differential, tangential gradient and tangential divergence. The third chapter is devoted to the topologies on the set of domains in R^{d}. Three topologies induced by convergence of domains are presented namely the convergence of charac- teristic functions, the convergence in the sense of Hausdorff and the convergence in the sense of compacts as well as the relationship between those different topologies. In the fourth chapter we present a shape optimization problem governed by linear state equations. After dealing with the continuity of the solution of the Laplacian problem with respect to the domain variation (including counter-examples to the continuity and the introduction to a new topology: the gamma-convergence), we analyse the existence of optimal shapes and the necessary condition of optimality in the case where an optimal shape exists. The shape optimization problems governed by nonlinear state equations are treated in chapter ve. The plan of study is the same as in chapter four that is continuity with respect to the domain variation of the solution of the p-Laplacian problem (and more general operator in divergence form), the existence of optimal shapes and the necessary condition of optimality in the case where an optimal shape exists. The last chapter deals with asymptotical shapes. After recalling the notion of Gamma-convergence, we study the asymptotic of the compliance functional in different situations. First we study the asymptotic of an optimal p-compliance-networks which is the compliance associated to p-Laplacian problem with control variables running in the class of one dimensional closed connected sets with assigned length. We provide also the connection with other asymptotic problems like the average distance problem. The asymptotic of the p-compliance-location which deal with the compliance associ- atied to the p-Laplacian problem with control variables running in the class of sets of finite numbers of points, is deduced from the study of the asymptotic of p-compliance- networks. Secondly we study the asymptotic of an optimal compliance-location. In this case we deal with the compliance associated to the classical Laplacian problem and the class of control variables is the class of identics n balls with radius depending on n and with fixed capacity.