Regularity of spectral fractional Dirichlet and Neumann problems

By Gerd Grubb. 2014

92 citations recorded by the source corpus.

Consider the fractional powers (ADir)a and (ANeu)a of the Dirichlet and Neumann realizations of a second‐order strongly elliptic differential operator A on a smooth bounded subset Ω of Rn . Recalling the results on complex powers and complex interpolation of domains of elliptic boundary value problems by Seeley in the 1970's, we demonstrate how they imply regularity properties in full scales of Hps ‐Sobolev spaces and Hölder spaces, for the solutions of the associated equations. Extensions to nonsmooth situations for low values of s are derived by use of recent results on H∞ ‐calculus. We also include an overview of the various Dirichlet‐ and Neumann‐type boundary problems associated with the fractional Laplacian.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/bd8fcdac738003c2663edb4c0c3a688231fd1c2c

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

Fractional elliptic equations, Caccioppoli estimates and regularity (2014, background): “rem 3.5]. When the coecients A(x) and the domain are smooth, the Lp{domain and regularity of fractional powers of strongly elliptic operators was considered in [27, 28], see also the recent preprint [17]1. For the fractional Laplacian on Rnwe know that the unique solution u2Hs(Rn) of the Dirichlet p”
What is the fractional Laplacian? A comparative review with new results (2018, background): “These spaces are discussed in [73,64], but we have more closely followed the exposition and notation of [18].”