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arXiv · 1707.03603

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

Nicola Abatangelo·Sven Jarohs·Alberto Saldaña
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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.

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Nicola Abatangelo, Sven Jarohs, Alberto Saldaña. 2017-07-12. Integral representation of solutions to higher-order fractional Dirichlet problems on balls. https://doi.org/10.1142/s0219199718500025

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