The thesis studies linear and semilinear Dirichlet problems driven by different fractional Laplacians. The boundary data can be smooth functions or also Radon measures. The goal is to classify the solutions which have a singularity on the boundary of the prescribed domain. We first remark the existence of a large class of harmonic functions with a boundary blow-up and we characterize them in terms of a new notion of degenerate boundary trace. Via some integration by parts formula, we then provide a weak theory of Stampacchia's sort to extend the linear theory to a setting including these functions: we study the classical questions of existence, uniqueness, continuous dependence on the data, regularity and asymptotic behaviour at the boundary. Afterwards we develop the theory of semilinear problems, by adapting and generalizing some sub- and supersolution methods. This allows us to build the fractional counterpart of large solutions in the elliptic PDE theory of nonlinear equations, giving sufficient conditions for the existence. The thesis is concluded with the definition and the study of a notion of nonlocal directional curvatures.

Large Solutions for Fractional Laplacian Operators

ABATANGELO, NICOLA
2015

Abstract

The thesis studies linear and semilinear Dirichlet problems driven by different fractional Laplacians. The boundary data can be smooth functions or also Radon measures. The goal is to classify the solutions which have a singularity on the boundary of the prescribed domain. We first remark the existence of a large class of harmonic functions with a boundary blow-up and we characterize them in terms of a new notion of degenerate boundary trace. Via some integration by parts formula, we then provide a weak theory of Stampacchia's sort to extend the linear theory to a setting including these functions: we study the classical questions of existence, uniqueness, continuous dependence on the data, regularity and asymptotic behaviour at the boundary. Afterwards we develop the theory of semilinear problems, by adapting and generalizing some sub- and supersolution methods. This allows us to build the fractional counterpart of large solutions in the elliptic PDE theory of nonlinear equations, giving sufficient conditions for the existence. The thesis is concluded with the definition and the study of a notion of nonlocal directional curvatures.
28-set-2015
Inglese
fractional Laplacian; nonlocal operators; large solutions; L1 weak solutions; nonlinear elliptic equations; Dirichlet problem; boundary singularity; nonlocal curvatures
VALDINOCI, ENRICO
VAN GEEMEN, LAMBERTUS
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/20.500.14242/86075
Il codice NBN di questa tesi è URN:NBN:IT:UNIMI-86075