# Extension theory and Kreĭn-type resolvent formulas for nonsmooth boundary value problems

@article{Abels2010ExtensionTA,
title={Extension theory and Kreĭn-type resolvent formulas for nonsmooth boundary value problems},
author={Helmut Abels and Gerd Grubb and Ian Wood},
journal={Journal of Functional Analysis},
year={2010},
volume={266},
pages={4037-4100}
}
• Published 19 August 2010
• Mathematics
• Journal of Functional Analysis
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## References

SHOWING 1-10 OF 80 REFERENCES
Krein resolvent formulas for elliptic boundary problems in nonsmooth domains
The paper reports on a recent construction of M-functions and Krein resolvent formulas for general closed extensions of an adjoint pair, and their implementation to boundary value problems for
Boundary triplets and M‐functions for non‐selfadjoint operators, with applications to elliptic PDEs and block operator matrices
• Mathematics
• 2008
Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl M‐function of extensions of the operators. The extensions are determined
Reduced and Generalized Stokes Resolvent Equations in Asymptotically Flat Layers, Part II: H∞-Calculus
Abstract.We study the generalized Stokes equations in asymptotically flat layers, which can be considered as compact perturbations of an infinite (flat) layer \$\Omega _0 = \mathbb{R}^{n - 1} \times (
Spectral Asymptotics for Nonsmooth Singular Green Operators
Singular Green operators G appear typically as boundary correction terms in resolvents for elliptic boundary value problems on a domain Ω ⊂ ℝ n , and more generally they appear in the calculus of
On coerciveness and semiboundedness of general boundary problems
The paper treats coerciveness inequalities (of the form Re(Au, u)≧c ‖u‖s2 −λ ‖u‖02,c>0,λ ∈ R) and semiboundedness inequalities (of the form Re (Au, u)≧−λ ‖u‖2) for the general boundary problems
Spectral theory of elliptic operators in exterior domains
AbstractDiverse closed (and selfadjoint) realizations of elliptic differential expressions A = Σ0⩽|α|,|β|⩽m(−1)αDαaα,β(x)Dβ, aα,β(·) ∈ C∞($$\bar \Omega$$) on smooth (bounded or unbounded) domains
Pseudodifferential Boundary Value Problems with Non-Smooth Coefficients
ABSTRACT In this contribution, we establish a calculus of pseudodifferential boundary value problems with Hölder continuous coefficients. It is a generalization of the calculus of pseudodifferential
A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains
• Mathematics
• 2009
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
On Stokes operators with variable viscosity in bounded and unbounded domains
• Mathematics
• 2009
We consider a generalization of the Stokes resolvent equation, where the constant viscosity is replaced by a general given positive function. Such a system arises in many situations as linearized
Properties of normal boundary problems for elliptic even-order systems
THEORY 1. The general set-up. The study of extensions of linear operators in Hilbert space is a well known tool in the theory of boundary value problems. The basic notions for the framework used here