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 e N. 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.

Abatangelo N., Jarohs S., Saldana A. (2018). Integral representation of solutions to higher-order fractional Dirichlet problems on balls. COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, 20(8), 1850002-1850037 [10.1142/S0219199718500025].

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

Abatangelo N.;
2018

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 e N. 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.
2018
Abatangelo N., Jarohs S., Saldana A. (2018). Integral representation of solutions to higher-order fractional Dirichlet problems on balls. COMMUNICATIONS IN CONTEMPORARY MATHEMATICS, 20(8), 1850002-1850037 [10.1142/S0219199718500025].
Abatangelo N.; Jarohs S.; Saldana A.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/835028
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