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The Laplacian on Cylindrical Domains

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Abstract

The aim of this paper is to prove elliptic regularity and parabolic maximal regularity of the Laplacian with mixed boundary conditions on domains Ω carrying a cylindrical structure. More precisely, we consider Ω to be given as the Cartesian product of whole or half spaces, a cube \({\mathcal{Q}}\) , and a standard domain V having compact boundary. Taking advantage of this structure we apply operator-valued Fourier multiplier results to transfer \({\mathcal{H}^{\infty}}\) -calculus results known for the Laplacian in L p(V) to the Laplacian in L p(Ω). This approach turns out to inherit elliptic regularity, i.e. the domain of the Dirichlet Laplacian equals \({W^{2,p}(\Omega) \cap W_0^{1,p}(\Omega)}\) , for instance. This is surprising since Ω may be unbounded and non-convex with boundary neither compact nor of class C 1,1 at the same time. More generally, we consider the following mixture of boundary conditions: on every smooth part of the boundary Dirichlet or Neumann boundary conditions are imposed and on parts related to \({\mathcal{Q}}\) generalized periodic boundary conditions are included. Via \({\mathcal{R}}\) -sectoriality we deduce maximal regularity in the parabolic sense which seems to be new for this general class of boundary conditions. Parabolic equations with such a mixture of boundary conditions on such type of domains appear for example in models describing growth of biological cells.

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  1. Institut für Angewandte Analysis, Universität Ulm, 89069, Ulm, Germany

    Tobias Nau

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  1. Tobias Nau
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Nau, T. The Laplacian on Cylindrical Domains. Integr. Equ. Oper. Theory 75, 409–431 (2013). https://doi.org/10.1007/s00020-012-2031-3

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