# 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} }

## 38 Citations

EXTENSION THEORY FOR ELLIPTIC PARTIAL DIFFERENTIAL OPERATORS WITH PSEUDODIFFERENTIAL METHODS

- Mathematics
- 2010

This is a short survey on the connection between general extension theories and the study of realizations of elliptic operators A on smooth domains in R n , n � 2. The theory of pseudodifferential…

The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates

- Mathematics
- 2011

Regularity of spectral fractional Dirichlet and Neumann problems

- Mathematics
- 2016

Consider the fractional powers (ADir)a and (ANeu)a of the Dirichlet and Neumann realizations of a second‐order strongly elliptic differential operator A on a smooth bounded subset Ω of Rn . Recalling…

Spectral Asymptotics for Nonsmooth Singular Green Operators

- Mathematics
- 2012

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…

Nonsmooth pseudodifferential boundary value problems on manifolds

- MathematicsJournal of Pseudo-Differential Operators and Applications
- 2018

We study pseudodifferential boundary value problems in the context of the Boutet de Monvel calculus or Green operators, with nonsmooth coefficients on smooth compact manifolds with boundary. In order…

Elliptic differential operators on Lipschitz domains and abstract boundary value problems

- MathematicsJournal of functional analysis
- 2014

Inverse problems with partial data for elliptic operators on unbounded Lipschitz domains

- MathematicsInverse Problems
- 2020

For a second order formally symmetric elliptic differential expression we show that the knowledge of the Dirichlet-to-Neumann map or Robin-to-Dirichlet map for suitably many energies on an…

Krein-like extensions and the lower boundedness problem for elliptic operators on exterior domains

- Mathematics
- 2010

δ′-Potentials Supported on Hypersurfaces

- Mathematics
- 2012

Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are…

SPECTRAL ASYMPTOTICS FOR ROBIN PROBLEMS WITH A DISCONTINUOUS COEFFICIENT

- Mathematics
- 2011

The spectral behavior of the difference between the resolvents of two realizations z A1 and z A2 of a second-order strongly elliptic symmetric differential operator A ,d ef ined by different Robin…

## References

SHOWING 1-10 OF 80 REFERENCES

Krein resolvent formulas for elliptic boundary problems in nonsmooth domains

- Mathematics
- 2008

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

- Mathematics
- 2005

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

- Mathematics
- 2012

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

- Mathematics
- 1971

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

- Mathematics
- 2008

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

- Mathematics
- 2005

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

- Mathematics
- 1974

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…