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Zbl 1001.53017
Kuwae, Kazuhiro; Machigashira, Yoshiroh; Shioya, Takashi
Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces. (English)
[J] Math. Z. 238, No.2, 269-316 (2001). ISSN 0025-5874; ISSN 1432-1823/e

This paper is about analysis on Alexandrov spaces. Earlier work of Otsu and the third of these authors demonstrated the existence of a $C^{1}$-differentiable structure on the set of nonsingular points. This allows one to define $L^{p}$ and $W^{1,p}.$ The first main result here is that if $\Omega $ is relatively compact, then $( W^{1,2}_{0} (\Omega) \subset L^{2} (\Omega))$ is compact. They define a Dirichlet form on $( W^{1,2}_{0} (\Omega) \times W^{1,2}_{0}(\Omega))$ and a corresponding Laplacian. This Laplacian has discrete spectrum and its eigenfunctions form a complete basis for $W^{1,2}_{0} (\Omega) $ and $L^{2} (\Omega).$ The existence of a heat kernel is proved and it is shown that solutions to the heat equation and eigenfunctions of the Laplacian are locally Hölder continuous.
[Thalia D.Jeffres (Morelia)]

MSC 2000:: *53C20 Riemannian manifolds (global)
53C70 Direct methods (G-spaces of Busemann, etc.)
58J35 Heat and other parabolic equation methods
58J50 Spectral problems; spectral geometry; scattering theory

Keywords: Alexandrov spaces; $C^{1}$-differentiable structure; Laplacian; eigenfunctions; heat kernel

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