A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains
By Fritz Gesztesy, M. Mitrea. 2009
120 citations recorded by the source corpus.
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 L2(Ω; dnx). 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 C1,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.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/1a1597ea91ca2d4ae3063108c563f8362ab9ff73
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Citations in the corpus
- Boundary pairs associated with quadratic forms (2012, methodology): “[66] M. Mitrea and M. Taylor, Boundary layer methods for Lipschitz domains in Riemannian manifolds, J. Funct.”
- Dirichlet-to-Neumann maps on bounded Lipschitz domains (2014, methodology): “These facts and the assertions on the domains below are essentially consequences of the regularity results domAD ⊂ H 3/2(Ω) and domAN ⊂ H 3/2(Ω) due to Jerison and Kenig [JK1, JK2], and Gesztesy and Mitrea [GM1, GM2].”
- Boundary Value Problems, Weyl Functions, and Differential Operators (2020, background): “[431, 432] and [92, 115, 323, 324, 325, 326] for some recent closely related works dealing with Schrödinger operators on Lipschitz domains.”
- Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples (2012, unknown): “…see, e.g. [Derkach and Malamud, 1991; Gesztesy, Makarov and Tsekanovskii, 1998; Krĕın, 1946; Langer and Textorius, 1977; Pankrashkin, 2006; Saakjan, 1965] for the general case and, e.g. [Gesztesy and Mitrea, 2008, 2009, 2011; Grubb, 2008; Posilicano and Raimondi, 2009] for applications to PDEs.”