This thesis is centred on nonlocal variational models as well as the study of the associated
function spaces and their applications. We focus in detail on two different but related nonlocal
models.
The first one is linked to image denoising, which is a core problem in image processing.
It consists of finding an approximation of a distorted image which is at the same time more
regular than the initial one. The competition between these two features endows the problem
with a natural variational structure, which gave rise to a broad investigation of minimization
based denoising models. In particular, in the last years, more and more attention was given to
nonlocal ones. We first proposed a new nonlocal model and analysed its features. Then, some
of them were established by us also for a more general class of nonlocal denoising models.
The second one is connected to a problem of fundamental importance in physical applications: the study of the shape of liquid drops and crystals lying in equilibrium under the
action of a bulk potential energy induced by an external force field. In this setting, non-trivial
configurations may arise since the potential competes with the surface tension of the drop
(or crystal). In the general case of drops (or crystals) of arbitrary volume many questions
concerning the equilibrium shapes are open. In the small volume case, several answers were
given in a celebrated paper by Figalli and Maggi. A nonlocal model addressing this problem
was studied by Cesaroni and Novaga under either periodicity or coercivity assumptions on
the potential. We studied this nonlocal model for a confining potential in the small volume
regime.
The organisation of the thesis is the following.
In Chapter 1 we study the notion of nonlocal variation induced by a suitable (non-
integrable) kernel K and we provide a detailed analysis of the space of nonlocal BV functions
with finite total K -variation. We give special emphasis on compactness, Lusin-type estimates,
Sobolev embeddings and isoperimetric and monotonicity properties of the K -variation and
the associated K -perimeter. Finally, we deal with the theory of Cheeger sets in this nonlocal
setting.
In Chapter 2, we study a general total variation denoising model with weighted L^1 fidelity,
where the regularizing term is a nonlocal variation induced by a suitable (non-integrable)
kernel K , and the approximation term is given by the L^1 norm with respect to a non-singular
measure with positively lower-bounded L^∞ density. We discuss regularity of the level sets and
uniqueness of solutions, both for high and low values of the fidelity parameter. Finally, we
analyse in detail the fidelity of the model in the case of binary data given by the characteristic
functions of convex sets.
In Chapter 3, we study the aforementioned nonlocal model related to the shapes of liquid
drops, for a confining potential in the small volume regime. Precisely, under coercivity and
locally Lipschitz regularity assumptions on the potential, we first prove that minimizers of
our energy are uniformly close to the Wulff shape associated to our nonlocal surface tension
(i.e. the Euclidean ball) and then we improve their regularity. As a consequence, we are able
to prove the convexity of our minimizers in this regime. We also give a quantitative estimate
on the rate of convergence depending on the volume of the minimizers.
In Chapter 4, we consider the previous model from a different perspective. For a fixed
value of the volume, we are interested in understanding when critical stable configurations
of our energy are indeed (local) minimizers. To this purpose, we initially derive first and
second variation formulae for our energy and later we investigate in which topologies local
minimality is obtained.