Matricial Coupling and Equivalence After Extension

Integr. equ. oper. theory

Speck

Teixeira

2005

The main objective is the study of a class of boundary value problems in weak formulation where two boundary conditions are given on the halflines bordering the first quadrant that contain impedance data and oblique derivatives. The associated operators are reduced by matricial coupling relations to certain boundary pseudodifferential operators which can be analyzed in detail. Results are: Fredholm criteria, explicit construction of generalized inverses in Bessel potential spaces, eventually after normalization, and regularity results. Particular interest is devoted to the impedance problem and to the oblique derivative problem, as well. Mathematics Subject Classification (2000). Primary 35J25; Secondary 30E25, 47G30, 45E10, 47A53, 47A20.

“…(a) Due to the last two results, L and T are toplinear equivalent (i.e., they coincide up to composition with boundedly invertible operators) [1],…”

Section: Theorem 23 ([6]mentioning

confidence: 99%

“…Vol. 56 (2006) Mixed Boundary Value Problems for the Helmholtz Equation 3 All considerations will be carried out for BVPs of normal type 1 where the so-called pre-symbol σ 1 of B 1 satisfies…”

Section: Introductionmentioning

confidence: 99%

Mixed Boundary Value Problems for the Helmholtz Equation in a Quadrant

Integr. equ. oper. theory

Speck

Teixeira

2005

“…(3.21) In the asymmetric case, it is a consequence of the formula (3.11), which can be regarded as a direct generalization of (3.21). At the end, the second (necessity) part of Theorem 3.8 is not so evident, particularly the construction of a coupling relation from an equivalence after extension relation, see [4]. However the conclusion that is most important in our applications can be proved independently and more directly as follows.…”

Section: Theorem 38 (Bart-tsekanovsky 1991) Let S and T Be Bounded mentioning

confidence: 92%

Diffraction from Polygonal-conical Screens, an Operator Approach

Operator Theory, Operator Algebras and Applications

Duduchava

Speck

2014

Abstract. The aim of this work is to construct explicitly resolvent operators for a class of boundary value problems in diffraction theory. These are formulated as boundary value problems for the three-dimensional Helmholtz equation with Dirichlet or Neumann conditions on a plane screen of polynomial-conical form (including unbounded and multiplyconnected screens), in weak formulation. The method is based upon operator theoretical techniques in Hilbert spaces, such as the construction of matrical coupling relations and certain orthogonal projections, which represent new techniques in this area of applications. Various cross connections are exposed, particularly considering classical WienerHopf operators in Sobolev spaces as general Wiener-Hopf operators in Hilbert spaces and studying relations between the crucial operators in game. Former results are extended, particularly to multiply-connected screens.Mathematics Subject Classification (2010). Primary 78A45; Secondary 47G30.

“…In the present section, we would like to go a little further and prove that, under some not very restrictive conditions, W Φ,Ω is equivalent after extension [1] to W Ψ,R+ . I.e., more than in (2.1), we can choose E 1 and F 1 to be bounded invertible.…”

Section: Relations With Wiener-hopf Operatorsmentioning

confidence: 99%

“…In general, it is not sufficient to suppose that only one of the operators is generalized invertible or to replace this by the closedness of the images, see [1,Example 6].…”

Section: Remark 22mentioning

confidence: 99%

Regularity of convolution type operators with PC symbols in Bessel potential spaces over two finite intervals

Mathematische Nachrichten

2003

Convolution type operators acting between Bessel potential spaces defined on a union of two finite intervals are studied from the point of view of their regularity properties. The operators are assumed to have kernels with Fourier transforms in the class of piecewise continuous matrix functions which is a frequent situation in several problems from mathematical physics. Various Wiener-Hopf operators having the same regularity properties as the considered convolution type operators are presented. Moreover, besides a criterion for the Fredholm property, sufficient conditions for the invertibility of the operators in study are obtained. The approach is based on the consideration of relations between different kinds of operators. In this way, the transfer of regularity properties, between the related operators, appears as a natural consequence. An application to a wave diffraction problem with impedance conditions is illustrated.