Regularity of spectral fractional Dirichlet and Neumann problems

TY - JOUR

T1 - Regularity of spectral fractional Dirichlet and Neumann problems

AU - Grubb, Gerd

PY - 2016/5

Y1 - 2016/5

N2 - 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 Hs p -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.

AB - 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 Hs p -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.

U2 - 10.1002/mana.201500041

DO - 10.1002/mana.201500041

M3 - Journal article

SN - 0025-584X

VL - 289

SP - 831

EP - 844

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

IS - 7

ER -