Integral representation of solutions to higher-order fractional Dirichlet problems on balls

TL;DR

This paper derives explicit formulas for solutions to higher-order fractional Dirichlet problems on balls, unifying previous methods and introducing new boundary operators, with applications to characterizations and properties of fractional harmonic functions.

Contribution

It introduces a unified approach with explicit formulas and a new boundary operator for higher-order fractional Laplacian problems on balls.

Findings

Explicit solution formulas for nonhomogeneous Dirichlet problems.

A new higher-order boundary operator generalizing normal derivatives.

Applications include characterizations of s-harmonic functions and Green function examples.

Abstract

We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet problems on balls involving any positive power $s>0$ of the Laplacian. We are able to prescribe values outside the domain and boundary data of different orders using explicit Poisson-type kernels and a new notion of higher-order boundary operator, which recovers normal derivatives if $s$ is a natural number. Our results unify and generalize previous approaches in the study of polyharmonic operators and fractional Laplacians. As applications, we show a novel characterization of $s$-harmonic functions in terms of Martin kernels, a higher-order fractional Hopf Lemma, and examples of positive and sign-changing Green functions.