## Thesis etd-04232023-211355 |

Link copiato negli appunti

Thesis type

Tesi di dottorato di ricerca

Author

BESSAS, KONSTANTINOS

URN

etd-04232023-211355

Thesis title

Nonlocal Models in Calculus of Variations

Academic discipline

MAT/05

Course of study

MATEMATICA

Supervisors

**tutor**Prof. Novaga, Matteo

Keywords

- fractional perimeter
- image denoising
- liquid drop model
- nonlocal perimeter

Graduation session start date

02/05/2023

Availability

Full

Summary

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.

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.

File

Nome file | Dimensione |
---|---|

Bessas_A...hesis.pdf | 93.71 Kb |

Bessas_P...hesis.pdf | 1.76 Mb |

Contatta l’autore |