In this thesis, we deal with problems related to nonlocal operators, in particular to the fractional Laplacian and some other types of fractional derivatives. We make an extensive introduction to the fractional Laplacian and to some related contemporary research themes. We add to this some original material: the potential theory of this operator and a proof of Schauder estimates with the potential theory approach, the study of a fractional elliptic problem in $mathbb{R}^n$ with convex nonlinearities and critical growth, and a stickiness property of $s$-minimal surfaces as $s$ gets small. Also, focusing our attention on some particular traits of the fractional Laplacian, we prove that other fractional operators have a similar behavior: Caputo stationary functions satisfy a particular density property in the space of smooth functions; an extension operator can be build for Marchaud-stationary functions.

SOME NONLOCAL OPERATORS AND EFFECTS DUE TO NONLOCALITY

BUCUR, CLAUDIA DALIA
2017

Abstract

In this thesis, we deal with problems related to nonlocal operators, in particular to the fractional Laplacian and some other types of fractional derivatives. We make an extensive introduction to the fractional Laplacian and to some related contemporary research themes. We add to this some original material: the potential theory of this operator and a proof of Schauder estimates with the potential theory approach, the study of a fractional elliptic problem in $mathbb{R}^n$ with convex nonlinearities and critical growth, and a stickiness property of $s$-minimal surfaces as $s$ gets small. Also, focusing our attention on some particular traits of the fractional Laplacian, we prove that other fractional operators have a similar behavior: Caputo stationary functions satisfy a particular density property in the space of smooth functions; an extension operator can be build for Marchaud-stationary functions.
20-apr-2017
Inglese
Fractional Laplacian; nonlocal operators; nonlocal minimal surfaces; fractional derivative; Caputo; Marchaud; Schauder estimates
VALDINOCI, ENRICO
MASTROPIETRO, VIERI
Università degli Studi di Milano
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.14242/80479
Il codice NBN di questa tesi è URN:NBN:IT:UNIMI-80479