 09111 Fritz Gesztesy and Marius Mitrea
 SelfAdjoint Extensions of the Laplacian and KreinType Resolvent
Formulas in Nonsmooth Domains
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Jul 12, 09

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Abstract. This paper has two main goals. First, we are concerned with the classification
of selfadjoint extensions of the Laplacian
$\Delta\big_{C^\infty_0(\Omega)}$ in
$L^2(\Omega; d^n x)$. Here, the domain
$\Omega$ belongs to a subclass of bounded Lipschitz domains (which we
term quasiconvex domains), which contain all convex domains, as well as
all domains of class $C^{1,r}$, for $r\in(1/2,1)$. Second, we establish
Kreintype formulas for the resolvents of the various selfadjoint
extensions of the Laplacian in quasiconvex domains and study the
properties of the corresponding WeylTitchmarsh operators (or energydependent
DirichlettoNeumann maps).
One significant technical innovation in this paper is an extension of
the classical
boundary trace theory for functions in spaces which lack Sobolev
regularity in a traditional sense, but are suitably adapted to the Laplacian.
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