Fractional-order operators: Boundary problems, heat equations

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The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential methods. The second half takes up the associated heat equation with homogeneous Dirichlet condition. Here we recall recently shown sharp results on interior regularity and on Lp-estimates up to the boundary, as well as recent Hölder estimates. This is supplied with new higher regularity estimates in L2 -spaces using a technique of Lions and Magenes, and higher Lp-regularity estimates (with arbitrarily high Hölder estimates in the time-parameter) based on a general result of Amann. Moreover, it is shown that an improvement to spatial C-regularity at the boundary is not in general possible.

Original languageEnglish
Title of host publicationMathematical Analysis and Applications-Plenary Lectures - ISAAC 2017
EditorsJoachim Toft, Luigi G. Rodino
Number of pages31
PublisherSpringer
Publication date2018
Pages51-81
ISBN (Print)9783030008734
DOIs
Publication statusPublished - 2018
Event11th International Society for Analysis, its Applications and Computation, ISAAC 2017 - Vaxjo, Sweden
Duration: 14 Aug 201718 Aug 2017

Conference

Conference11th International Society for Analysis, its Applications and Computation, ISAAC 2017
LandSweden
ByVaxjo
Periode14/08/201718/08/2017
SeriesSpringer Proceedings in Mathematics & Statistics
Volume262
ISSN2194-1009

    Research areas

  • Dirichlet and Neumann conditions, Fractional Laplacian, Green’s formula, Heat equation, Pseudodifferential operator, Space-time regularity, Stable process

ID: 214023447