Large solutions for fractional Laplacian operators

TL;DR

This thesis explores solutions to fractional Laplacian problems with boundary singularities, establishing existence, uniqueness, and regularity results, and introduces a new boundary trace concept and nonlocal curvatures.

Contribution

It introduces a new boundary trace notion for fractional Laplacians, extends linear theory to include boundary blow-up solutions, and develops a fractional large solutions theory.

Findings

Existence of harmonic functions with boundary blow-up

Characterization of boundary blow-up solutions via degenerate boundary trace

Development of a fractional large solutions theory

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.