## Thesis etd-07242017-170705 |

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Thesis type

Elaborati finali per laurea triennale

Author

PICENNI, NICOLA

URN

etd-07242017-170705

Thesis title

Nonlocal characterizations of Sobolev spaces and functions of bounded variation in dimension one

Department

MATEMATICA

Course of study

MATEMATICA

Supervisors

**relatore**Prof. Gobbino, Massimo

Keywords

- Sobolev spaces
- functions of bounded variation
- gamma-convergence
- nonlocal functionals

Graduation session start date

14/07/2017

Availability

Full

Summary

In the thesis we study two kinds of families of nonlocal functionals, whose limits (in the

appropriate sense) are multiples either of the Sobolev norm or of the total variation,

depending on the value of a parameter p.

This provides a characterization of the Sobolev spaces and of the space of

functions of bounded variation as the set of functions for which the (Γ-)limit of those families is finite.

For the sake of simplicity, we consider only the one dimensional case, even if almost

all the results are valid in every dimension.

In the first family, derivatives are replaced by finite differences weighted by a family of mollifiers. As shown in 2001 by J. Bourgain, H. Brezis and P. Mironescu, this family converges to the Sobolev norm or the total variation both in the sense of pointwise convergence and in the sense of De Giorgi’s Γ-convergence.

The second family has been studied in a series of paper (Nguyen (JFA 2006, Duke 2011), Bourgain-Nguyen (CRAS 2006), Brezis-Nguyen (ArXiv 2016)). In these papers the authors proved that (if p>1) the pointwise limit is equal to a multiple of the Sobolev norm, and that the Γ-limit is also a multiple of the Sobolev norm (or the total variation if p=1), but with a strictly lower constant.

Nevertheless, many problems remained open.

In this thesis we answer two of them:

• we compute the exact value of the constant that appears in the Γ-limit (that was only conjectured until now),

• we show the existence of smooth recovery families.

The techniques we use provide also a simpler proof of the Γ-convergence result.

Some parts of this thesis are based on a joint work with C. Antonucci, M. Gobbino and M. Migliorini.

appropriate sense) are multiples either of the Sobolev norm or of the total variation,

depending on the value of a parameter p.

This provides a characterization of the Sobolev spaces and of the space of

functions of bounded variation as the set of functions for which the (Γ-)limit of those families is finite.

For the sake of simplicity, we consider only the one dimensional case, even if almost

all the results are valid in every dimension.

In the first family, derivatives are replaced by finite differences weighted by a family of mollifiers. As shown in 2001 by J. Bourgain, H. Brezis and P. Mironescu, this family converges to the Sobolev norm or the total variation both in the sense of pointwise convergence and in the sense of De Giorgi’s Γ-convergence.

The second family has been studied in a series of paper (Nguyen (JFA 2006, Duke 2011), Bourgain-Nguyen (CRAS 2006), Brezis-Nguyen (ArXiv 2016)). In these papers the authors proved that (if p>1) the pointwise limit is equal to a multiple of the Sobolev norm, and that the Γ-limit is also a multiple of the Sobolev norm (or the total variation if p=1), but with a strictly lower constant.

Nevertheless, many problems remained open.

In this thesis we answer two of them:

• we compute the exact value of the constant that appears in the Γ-limit (that was only conjectured until now),

• we show the existence of smooth recovery families.

The techniques we use provide also a simpler proof of the Γ-convergence result.

Some parts of this thesis are based on a joint work with C. Antonucci, M. Gobbino and M. Migliorini.

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