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

Abels, Helmut; Grubb, Gerd; Wood, Ian

doi:10.1016/j.jfa.2014.01.016

Cited by 34 publications

(122 citation statements)

References 43 publications

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“…For nonsmooth cases, we remark that there do exist pseudodifferential theories requiring only limited smoothness in x, cf. [2] and other works of Abels listed there. Applications to the present problems await development.…”

Section: Nonsmooth Situationsmentioning

confidence: 94%

Regularity of spectral fractional Dirichlet and Neumann problems

Grubb

2015

Mathematische Nachrichten

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Received XXXX, revised XXXX, accepted XXXX Published online XXXX MSC (2000) 35P99, 35S15, 47G30Consider the fractional powers (ADir) a and (ANeu) a of the Dirichlet and Neumann realizations of a secondorder strongly elliptic differential operator A on a smooth bounded subset Ω of R n . Recalling the results on complex powers and complex interpolation of domains of elliptic boundary value problems by Seeley in the 1970's, we demonstrate how they imply regularity properties in full scales of H s p -Sobolev spaces and Hölder spaces, for the solutions of the associated equations. Extensions to nonsmooth situations for low values of s are derived by use of recent results on H ∞ -calculus. We also include an overview of the various Dirichlet-and Neumann-type boundary problems associated with the fractional Laplacian.

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Section: Nonsmooth Situationsmentioning

confidence: 94%

Regularity of spectral fractional Dirichlet and Neumann problems

Grubb

2015

Mathematische Nachrichten

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“…Finally, the sum is direct, since N 1 (z) ∩ H 1,D = {0} by the uniqueness of the weak Dirichlet solution, cf. (1).…”

Section: P R O O F (I) Assume That H 1 and H 2 Are Two Solutions Ofmentioning

confidence: 99%

“…He also applies his concept to Laplacians on open subsets

X \subset {double-struckR}^{n}

with smooth boundary, but the surjectivity condition on

Γ^{'}

enforces

G = {sans-serifH}^{1 / 2} (\partial X)

. Other results on Krein‐type resolvent formulae are also considered in , , , .…”

Section: Introductionmentioning

confidence: 99%

Boundary pairs associated with quadratic forms

Post

2015

Math. Nachr.

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We introduce a purely functional analytic framework for elliptic boundary value problems in a variational form. We define abstract Neumann and Dirichlet boundary conditions and a corresponding Dirichlet‐to‐Neumann operator, and develop a theory relating resolvents and spectra of these operators. We illustrate the theory by many examples including Jacobi operators, Laplacians on spaces with (non‐smooth) boundary, the Zaremba (mixed boundary conditions) problem and discrete Laplacians.

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“…Since that time the Weyl function became very important tool for ordinary differential operators. However, the application area of the Weyl function was permanently extended to other fields, for instance, to partial differential operators [14,12,34,55,35,1], to operators different from the Sturm-Liouville operator [48] and even to non-linear evolution equations [36,76,78].…”

Section: Weyl Functions and Spectra Of Proper Extensionsmentioning

confidence: 99%

“…Other versions of the Krein-type formula as well as its applications to boundary value problems can also be found in [34,35,1,65,13]. The proof of the following result relies on formula (2.5).…”

Section: Kreȋn-type Formula For Resolvents and Comparabilitymentioning

confidence: 99%

Trace formulas for additive and non-additive perturbations

Malamud

Neidhardt

2015

Advances in Mathematics

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Trace formulas for pairs of self-adjoint, maximal dissipative and accumulative as well as other types of resolvent comparable operators are obtained. In particular, the existence of a complex-valued spectral shift function for a pair {H , H} of maximal accumulative operators has been proved. We investigate also the existence of a real-valued spectral shift function. Moreover, we treat in detail the case of additive trace class perturbations. Assuming that H and H = H +V are maximal accumulative and V is trace class, we prove the existence of a summable complex-valued spectral shift function. We also obtain trace formulas for pairs {H, H * } assuming only that H and H * are resolvent comparable. In this case the determinant of the characteristic function of H is involved in trace formulas. The results rely on the technique of boundary triplets for densely and non-densely defined closed symmetric operators. The approach allows to express the spectral shift function of a pair of extensions in terms of the abstract Weyl function and the boundary operator. We improve and generalize certain classical results of M.G. Kreȋn for pairs of self-adjoint and dissipative operators, results of A.V. Rybkin for such pairs as well as of

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

Cited by 34 publications

References 43 publications

Regularity of spectral fractional Dirichlet and Neumann problems

Regularity of spectral fractional Dirichlet and Neumann problems

Boundary pairs associated with quadratic forms

Trace formulas for additive and non-additive perturbations

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