Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator |
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Authors: | W. Arendt A. F. M. Ter Elst M. Warma |
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Affiliation: | 1. Institute of Applied Analysis, University of Ulm, Ulm, Germany;2. Department of Mathematics, University of Auckland, Auckland, New Zealand;3. Department of Mathematics, University of Puerto Rico (Rio Piedras Campus), College of Natural Sciences, San Juan, PR, USA |
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Abstract: | In the very influential paper [4 Caffarelli, L.A., Silvestre, L. (2007). An extension problem related to the fractional Laplacian. Commun. Partial Differential Equations 32:1245–1260.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] Caffarelli and Silvestre studied regularity of (?Δ)s, 0<s<1, by identifying fractional powers with a certain Dirichlet-to-Neumann operator. Stinga and Torrea [15 Stinga, P.R., Torrea, J. (2010). Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differential Equations 35:2092–2122.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] and Galé et al. [7 Galé, J., Miana, P., Stinga, P.R. (2013). Extension problem and fractional operators: semigroups and wave equations. J. Evol. Eqn. 13:343–368.[Crossref], [Web of Science ®] , [Google Scholar]] gave several more abstract versions of this extension procedure. The purpose of this paper is to study precise regularity properties of the Dirichlet and the Neumann problem in Hilbert spaces. Then the Dirichlet-to-Neumann operator becomes an isomorphism between interpolation spaces and its part in the underlying Hilbert space is exactly the fractional power. |
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Keywords: | Caffarelli–Silvestre extension Dirichlet and Neumann problem Dirichlet-to-Neumann operator fractional power maximal regularity of solutions sectorial operator |
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