Wave Factorization of Elliptic Symbols: Theory and Applications

Vasil’ev, V. B.

doi:10.1007/978-94-015-9448-6

Cited by 62 publications

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“…From the historical point of view, one can say, that the story started with the solution of Sommerfeld's half-plane problem [35] by modern WienerHopf methods [26], contributions to the diffraction by a quarter-plane [11,12,25,44] and the discovery of relations with general WHOs [31,37]. The present paper could be regarded as an extension of [24] to non-convex, general polynomial-conical screens, however with several new techniques that provide a deeper insight into the structure of this kind of BVPs.…”

Section: Formulation Of Problems and Main Resultsmentioning

confidence: 99%

“…Vasil'ev proposed in his book [44] to solve the diffraction problem by use of a so-called wave factorization of the function t(ξ) = (ξ 2 − k 2 ) 1/2 into two factors, holomorphic in certain tube domains. However, looking at the explicit form of the two factors, it turns out that they vanish within the corresponding tube domains, see [44], pages 28-29 and 38-39. This means that the given factorization is not a wave factorization in the sense of the author's own Definition 5.1 and therefore not helpful for the solution of the problem.…”

Section: Explicit Solution Of the Bvpsmentioning

confidence: 99%

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Diffraction from Polygonal-conical Screens, an Operator Approach

Castro

Duduchava

Speck

2014

Operator Theory, Operator Algebras and Applications

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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.

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Section: Formulation Of Problems and Main Resultsmentioning

confidence: 99%

Section: Explicit Solution Of the Bvpsmentioning

confidence: 99%

Diffraction from Polygonal-conical Screens, an Operator Approach

Castro

Duduchava

Speck

2014

Operator Theory, Operator Algebras and Applications

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“…the so-called homogeneous wave factorization [15]. Let us note that this case is more pleasant for consideration because we can reduce the system (5.1) to a system of linear algebraic equations by the Mellin transform [15]. …”

Section: The Homogeneous Wave Factorizationmentioning

confidence: 99%

“…But for a simplest domain with a non-smooth boundary the model domain is a cone. The author supposes to consider a cone like a canonical domain serving the theory of pseudo-differential equations on manifolds with non-smooth boundary [15]. Such approach is based on a special representation for an elliptic symbol.…”

Section: Introductionmentioning

confidence: 99%

“…Such approach is based on a special representation for an elliptic symbol. Existence of such special wave factorization for symbols of elliptic pseudo differential equations has permitted to obtain full solvability picture for model pseudo differential equations in two-dimensional case [15,16]. Recently the author has found that there is another way to develop the theory [19][20][21].…”

Section: Introductionmentioning

confidence: 99%

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Pseudo-differential equations and conical potentials: 2-dimensional case

Vasilyev¹

2019

Opuscula Math.

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Abstract. We consider two-dimensional elliptic pseudo-differential equation in a plane sector. Using a special representation for an elliptic symbol and the formula for a general solution we study the Dirichlet problem for such equation. This problem was reduced to a system of linear integral equations and then after some transformations to a system of linear algebraic equations. The unique solvability for the Dirichlet problem was proved in Sobolev-Slobodetskii spaces and a priori estimate for the solution is given.

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Pseudodifferential equations, wave factorization, and related problems

Vasilyev

2018

Math Methods in App Sciences

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We consider a model elliptic pseudodifferential equation in a special canonical domains of a multidimensional space. Using a special representation for an elliptic symbol, we give the formula for a general solution of such an equation and choose additional conditions under which this boundary value problem has a unique solution in appropriate Sobolev-Slobodetskii spaces. Also, we introduce some transmutation operators that help us in constructing the solution.

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Wave Factorization of Elliptic Symbols: Theory and Applications

Cited by 62 publications

References 0 publications

Diffraction from Polygonal-conical Screens, an Operator Approach

Diffraction from Polygonal-conical Screens, an Operator Approach

Pseudo-differential equations and conical potentials: 2-dimensional case

Pseudodifferential equations, wave factorization, and related problems

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