In this thesis, we deal with problems related to nonlocal operators, in particular to the fractional Laplacian and to some other types of fractional derivatives (the Caputo and the Marchaud derivatives). We make an extensive introduction to the fractional Laplacian, we present some related contemporary research results and we add some original material. Indeed, we study the potential theory of this operator, introduce a new proof of Schauder estimates using the potential theory approach, we study a fractional elliptic problem in Rn with convex nonlinearities and critical growth and we present a stickiness property of nonlocal minimal surfaces for small values of the fractional parameter. Also, we point out that the (nonlocal) character of the fractional Laplacian gives rise to some surprising nonlocal effects. We prove that other fractional operators have a similar behavior: in particular, Caputo-stationary functions are dense in the space of smooth functions; moreover, we introduce an extension operator for Marchaud-stationary functions.

(2017). Some nonlocal operators and effects due to nonlocality. (Tesi di dottorato, Università degli Studi di Milano, 2017).

Some nonlocal operators and effects due to nonlocality

BUCUR, CLAUDIA DALIA
2017-04-20

Abstract

In this thesis, we deal with problems related to nonlocal operators, in particular to the fractional Laplacian and to some other types of fractional derivatives (the Caputo and the Marchaud derivatives). We make an extensive introduction to the fractional Laplacian, we present some related contemporary research results and we add some original material. Indeed, we study the potential theory of this operator, introduce a new proof of Schauder estimates using the potential theory approach, we study a fractional elliptic problem in Rn with convex nonlinearities and critical growth and we present a stickiness property of nonlocal minimal surfaces for small values of the fractional parameter. Also, we point out that the (nonlocal) character of the fractional Laplacian gives rise to some surprising nonlocal effects. We prove that other fractional operators have a similar behavior: in particular, Caputo-stationary functions are dense in the space of smooth functions; moreover, we introduce an extension operator for Marchaud-stationary functions.
Enrico Valdinoci
nonlocal operators, fractional Laplacian, Caputo derivative, Marchaud derivative, nonlocal minimal surfaces
English
CORSO DI DOTTORATO IN MATEMATICA
XXIX
2017
Università degli Studi di Milano
(2017). Some nonlocal operators and effects due to nonlocality. (Tesi di dottorato, Università degli Studi di Milano, 2017).
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10281/277792
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