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 / N. Abatangelo ; tutor: E. Valdinoci; coordinatore: L. van Geemen. DIPARTIMENTO DI MATEMATICA "FEDERIGO ENRIQUES", 2015 Sep 28. 28. ciclo, Anno Accademico 2015. [10.13130/abatangelo-nicola_phd2015-09-28].

Large Solutions for Fractional Laplacian Operators

N. Abatangelo
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
Settore MAT/05 - Analisi Matematica
fractional Laplacian; nonlocal operators; large solutions; L1 weak solutions; nonlinear elliptic equations; Dirichlet problem; boundary singularity; nonlocal curvatures
VALDINOCI, ENRICO
VAN GEEMEN, LAMBERTUS
Doctoral Thesis
Large Solutions for Fractional Laplacian Operators / N. Abatangelo ; tutor: E. Valdinoci; coordinatore: L. van Geemen. DIPARTIMENTO DI MATEMATICA "FEDERIGO ENRIQUES", 2015 Sep 28. 28. ciclo, Anno Accademico 2015. [10.13130/abatangelo-nicola_phd2015-09-28].
File in questo prodotto:
File Dimensione Formato  
phd_unimi_R10392.pdf

accesso aperto

Tipologia: Tesi di dottorato completa
Dimensione 1.62 MB
Formato Adobe PDF
1.62 MB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/320258
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact