Fractional-order operators: Boundary problems, heat equations

Gerd Grubb

doi:10.1007/978-3-030-00874-1_2

Fractional-order operators: Boundary problems, heat equations

Gerd Grubb^*

^*Corresponding author for this work

Department of Mathematical Sciences

6 Citations (Scopus)

Abstract

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 L_p-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 L_p-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 language	English
Title of host publication	Mathematical Analysis and Applications-Plenary Lectures - ISAAC 2017
Editors	Joachim Toft, Luigi G. Rodino
Number of pages	31
Publisher	Springer
Publication date	2018
Pages	51-81
ISBN (Print)	9783030008734
DOIs	https://doi.org/10.1007/978-3-030-00874-1_2
Publication status	Published - 2018
Event	11th International Society for Analysis, its Applications and Computation, ISAAC 2017 - Vaxjo, Sweden Duration: 14 Aug 2017 → 18 Aug 2017

Conference

Conference	11th International Society for Analysis, its Applications and Computation, ISAAC 2017
Country/Territory	Sweden
City	Vaxjo
Period	14/08/2017 → 18/08/2017

Series	Springer Proceedings in Mathematics & Statistics
Volume	262
ISSN	2194-1009

Keywords

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

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10.1007/978-3-030-00874-1_2

Cite this

Fractional-order operators: Boundary problems, heat equations. / Grubb, Gerd.
Mathematical Analysis and Applications-Plenary Lectures - ISAAC 2017. ed. / Joachim Toft; Luigi G. Rodino. Springer, 2018. p. 51-81 (Springer Proceedings in Mathematics & Statistics, Vol. 262).

Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review

Grubb, G 2018, Fractional-order operators: Boundary problems, heat equations. in J Toft & LG Rodino (eds), Mathematical Analysis and Applications-Plenary Lectures - ISAAC 2017. Springer, Springer Proceedings in Mathematics & Statistics, vol. 262, pp. 51-81, 11th International Society for Analysis, its Applications and Computation, ISAAC 2017, Vaxjo, Sweden, 14/08/2017. https://doi.org/10.1007/978-3-030-00874-1_2

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KW - Dirichlet and Neumann conditions

KW - Fractional Laplacian

KW - Green’s formula

KW - Heat equation

KW - Pseudodifferential operator

KW - Space-time regularity

KW - Stable process

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Fractional-order operators: Boundary problems, heat equations

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