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.
Integral representation of solutions to higher-order fractional Dirichlet problems on balls / Abatangelo N.; Jarohs S.; Saldana A.. - In: COMMUNICATIONS IN CONTEMPORARY MATHEMATICS. - ISSN 0219-1997. - STAMPA. - 20:8(2018), pp. 1850002.1850002-1850002.1850037. [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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.