Some nonlocal operators and effects due to nonlocality

TL;DR

This thesis explores nonlocal operators like the fractional Laplacian, revealing unique effects and introducing new theoretical tools, including Schauder estimates, and examining properties of fractional derivatives and nonlocal minimal surfaces.

Contribution

It provides new proofs of Schauder estimates, studies fractional elliptic problems with nonlinearities, and introduces novel concepts like the extension operator for Marchaud-stationary functions.

Findings

Fractional Laplacian exhibits surprising nonlocal effects.

Caputo-stationary functions are dense in smooth functions.

Extension operator for Marchaud-stationary functions is introduced.

Abstract

In this PhD 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 $\mathbb{R}^n$ 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.