Nonhomogeneous boundary conditions for the spectral fractional Laplacian

TL;DR

This paper develops a framework for solving boundary value problems involving the spectral fractional Laplacian with nonhomogeneous boundary conditions, including measure data and nonlinearities, using harmonic functions and weak-$L^1$ theory.

Contribution

It introduces a method to construct harmonic functions for the spectral fractional Laplacian and classifies their boundary divergence profiles, enabling the treatment of nonhomogeneous boundary conditions.

Findings

Established well-posedness for measure data boundary problems.

Developed a sub- and supersolution method for linear and semilinear cases.

Proved existence of large solutions for certain nonlinearities.

Abstract

We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their divergent profile at the boundary. This is used to establish and solve boundary value problems associated with nonhomogeneous boundary conditions. We provide a weak-$L^1$ theory to show how problems with measure data at the boundary and inside the domain are well-posed. We study linear and semilinear problems, performing a sub- and supersolution method, and we finally show the existence of large solutions for some power-like nonlinearities.