• Asymptotic shapes
• Compliance functional
• Gamma-convergence
• Location problems.
• Network problems
• p-Laplacian equation
• Stazionary configuration

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.