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A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains

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Journal d'Analyse Mathématique Aims and scope

Abstract

This paper has two main goals. First, we are concerned with a description of all self-adjoint extensions of the Laplacian \( - \Delta {|_{C_0^\infty (\Omega )}}\) in L 2(Ω; d n x). Here, the domain Ω belongs to a subclass of bounded Lipschitz domains (which we term quasi-convex domains), that contains all convex domains as well as all domains of class C 1,r, for r > 1/2. Second, we establish Kreĭn-type formulas for the resolvents of the various self-adjoint extensions of the Laplacian in quasiconvex domains and study the well-posedness of boundary value problems for the Laplacian as well as basic properties of the corresponding Weyl-Titchmarsh operators (or energy-dependent Dirichlet-to-Neumann maps). One significant innovation in this paper is an extension of the classical boundary trace theory for functions in spaces that lack Sobolev regularity in a traditional sense, but are suitably adapted to the Laplacian.

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  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Fritz Gesztesy & Marius Mitrea

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  1. Fritz Gesztesy
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  2. Marius Mitrea
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Correspondence to Fritz Gesztesy.

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Dedicated to the memory of M. Sh. Birman (1928–2009)

Based upon work partially supported by the US National Science Foundation under Grant No. DMS-0400639

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Gesztesy, F., Mitrea, M. A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains. JAMA 113, 53–172 (2011). https://doi.org/10.1007/s11854-011-0002-2

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  • DOI: https://doi.org/10.1007/s11854-011-0002-2

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