Spectral asymptotics for Robin problems with a discontinuous coefficient

Grubb, Gerd

doi:10.4171/jst/7

Cited by 27 publications

(38 citation statements)

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“…In the finer asymptotic estimate (1.7)-(1.8), p 0 (x , ξ ) denotes the principal symbol of P γ ,ν andk 0 (x , ξ , ξ n ) is the principal symbol-kernel of K γ ; the derivation of the formula is explained in [31], Th. 2.4.…”

Section: Birman's Methods Revisitedmentioning

confidence: 99%

“…The estimate for A −1 χ − A −1 γ was later improved to an asymptotic estimate (in [24] and [9], the latter including exterior domains): (1. 8) this has been extended to nonsmooth b in [31] (the ingredients in the formula are explained around Theorem 2.4 there).…”

Section: (12)mentioning

confidence: 96%

“…To find the asymptotic behavior of the s-numbers we shall use the following theorem shown in [31] (Th. 3.3):…”

Section: Spectral Asymptoticsmentioning

confidence: 99%

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

Grubb

2011

Journal of Mathematical Analysis and Applications

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Keywords:Mixed boundary condition Zaremba problem Resolvent difference Dirichlet-to-Neumann operator Krein resolvent formula Spectral asymptotics Weak Schatten class Nonstandard pseudodifferential operator For a second-order symmetric strongly elliptic operator A on a smooth bounded open set in R n , the mixed problem is defined by a Neumann-type condition on a part Σ + of the boundary and a Dirichlet condition on the other part Σ − . We show a Kreȋn resolvent formula, where the difference between its resolvent and the Dirichlet resolvent is expressed in terms of operators acting on Sobolev spaces over Σ + . This is used to obtain a new Weyltype spectral asymptotics formula for the resolvent difference (where upper estimates were known before), namely s j j 2/(n−1) → C 2/(n−1) 0,+ , where C 0,+ is proportional to the area of Σ + , in the case where A is principally equal to the Laplacian.The mixed boundary value problem for a second-order strongly elliptic symmetric operator A on a smooth bounded open set Ω ⊂ R n with boundary Σ , in case of the Laplacian also called the Zaremba problem, is defined by a Neumann-type condition on a part of the boundary Σ + and a Dirichlet condition on the other part Σ − . It does not have the regularity of standard elliptic boundary problems (the L 2 -domain is at best in H 3 2 −ε (Ω)). It has been analyzed with regards to regularity and mapping properties e.g. in Peetre [45,46], Shamir [52], Eskin [14], Pryde [48], Rempel and Schulze [49], Simanca [53], Harutyunyan and Schulze [33]. We shall here study it from the point of view of extension theory for elliptic operators. There has been a recent revival in the interest for connections between abstract extension theories for operators in Hilbert space (as initiated by Krein [36], Vishik [54], Birman [6], Grubb [22] and others) and interpretations to boundary value problems for partial differential operators. Cf. e.g. Amrein and Pearson [3], Pankrashkin [44], Behrndt and Langer [4], Ryzhov [50], Brown, Marletta, Naboko and Wood [13], Alpay and Behrndt [2], Malamud [41], based on boundary triples theory (as developed from the book of Gorbachuk and Gorbachuk [20] and its sources). Other methods are used in the works of Brown, Grubb and Wood [12,28], Posilicano and Raimondi [47], Gesztesy and Mitrea [16-18] (and their references); see also and Abels, Grubb and Wood [1]. One of the interesting aims has been to derive Kreȋn resolvent formulas that link the resolvent of a general operator with the resolvent of a fixed reference operator by expressing the difference in terms of operators connected to the boundary.For the mixed problem, a Kreȋn resolvent formula connecting the operator to the Dirichlet realization was worked out in [44], based on boundary triples theory. A different formula results from [22,24], see also [12], Sect. 3.2.5. Observations on the connection with the Neumann realization were given in [41]. An upper bound for the spectral behavior of the resolvent difference was shown by Birman in [6].In the present paper we sha...

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Section: Birman's Methods Revisitedmentioning

confidence: 99%

Section: (12)mentioning

confidence: 96%

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

Grubb

2011

Journal of Mathematical Analysis and Applications

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“…The simpler form of the boundary conditions presented in this article allows to relax the regularity condition imposed in Definition 3.14, as will be shown in Example 5.5, and treat discontinuities in smooth subsets of codimension one. Therefore generalizing the results appearing in [21].…”

Section: Relations To Existing Approachesmentioning

confidence: 94%

“…Also equivariant and Robin-type Laplacians can be naturally described in terms of closed and semi-bounded quadratic forms (see, e.g., [19,21,30,32]). In this context the subtle relation between quadratic forms and representing operators manifests through the fact that the form domain D(Q) always contains the operator domain D(T ) of the representing operator.…”

Section: Introductionmentioning

confidence: 99%

Self-adjoint extensions of the Laplace–Beltrami operator and unitaries at the boundary

Ibort

Lledó

Pérez-Pardo

2015

Journal of Functional Analysis

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We construct in this article a class of closed semi-bounded quadratic forms on the space of square integrable functions over a smooth Riemannian manifold with smooth compact boundary. Each of these quadratic forms specifies a semibounded self-adjoint extension of the Laplace-Beltrami operator. These quadratic forms are based on the Lagrange boundary form on the manifold and a family of domains parametrized by a suitable class of unitary operators on the boundary that will be called admissible. The corresponding quadratic forms are semi-bounded below and closable. Finally, the representing operators correspond to semi-bounded self-adjoint extensions of the Laplace-Beltrami operator. This family of extensions is compared with results existing in ✩ 635 Boundary conditions the literature and various examples and applications are discussed.

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Quasi Boundary Triples, Self-adjoint Extensions, and Robin Laplacians on the Half-space

Behrndt

Schlosser

2019

Linear Systems, Signal Processing and Hypercomplex Analysis

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In this note self-adjoint extensions of symmetric operators are investigated by using the abstract technique of quasi boundary triples and their Weyl functions. The main result is an extension of [5, Theorem 2.6] which provides sufficient conditions on the parameter in the boundary space to induce self-adjoint realizations. As an example self-adjoint Robin Laplacians on the half-space with boundary conditions involving an unbounded coefficient are considered. Quasi boundary triples and self-adjoint extensions

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Spectral asymptotics for Robin problems with a discontinuous coefficient

Cited by 27 publications

References 28 publications

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

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

Self-adjoint extensions of the Laplace–Beltrami operator and unitaries at the boundary

Quasi Boundary Triples, Self-adjoint Extensions, and Robin Laplacians on the Half-space

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