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

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.

arXiv comment34 pages, revised version

References in corpus (3)

• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
• Some observations on the Green function for the ball in the fractional Laplace framework
• On the maximum principle for higher-order fractional Laplacians

Cited by in corpus (7)

• On the maximum principle for higher-order fractional Laplacians
• Fractional-Order Operators on Nonsmooth Domains
• An asymptotic expansion for the fractional $p$-Laplacian and for gradient dependent nonlocal operators
• Optimal operator preconditioning for pseudodifferential boundary problems
• Exact Green's formula for the fractional Laplacian and perturbations
• Concentration phenomena for the fractional $Q$-curvature equation in dimension 3 and fractional Poisson formulas
• A Brezis and Peletier type result for the fractional Robin function