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Fractional-Order Operators: Boundary Problems, Heat Equations

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Mathematical Analysis and Applications—Plenary Lectures (ISAAC 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 262))

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 \(L_2\)-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 \(C^\infty \)-regularity at the boundary is not in general possible.

Expanded version of a lecture given at the ISAAC Conference August 2017, Växjö University, Sweden.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Copenhagen University, Universitetsparken 5, 2100, Copenhagen, Denmark

    Gerd Grubb

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  1. Gerd Grubb
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Correspondence to Gerd Grubb .

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Editors and Affiliations

  1. Dipartimento di Matematica, Università di Torino, Turin, Italy

    Luigi G. Rodino

  2. Department of Mathematics, Linnaeus University, Växjö, Sweden

    Joachim Toft

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Grubb, G. (2018). Fractional-Order Operators: Boundary Problems, Heat Equations. In: Rodino, L., Toft, J. (eds) Mathematical Analysis and Applications—Plenary Lectures. ISAAC 2017. Springer Proceedings in Mathematics & Statistics, vol 262. Springer, Cham. https://doi.org/10.1007/978-3-030-00874-1_2

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