Pseudodifferential equations, wave factorization, and related problems

Vasilyev, Vladimir

doi:10.1002/mma.5212

Cited by 20 publications

(32 citation statements)

References 14 publications

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“…We will use our constructions from [16,17] to describe the general solution of the equation ( 1) for the case ae…”

Section: A General Solutionmentioning

confidence: 99%

“…For first three cases 1, 2, 3 of Theorem 3 we can give an a priori estimate for the solution using the following lifting lemma; it was proved in [17].…”

Section: A Priori Estimatesmentioning

confidence: 99%

“…P Remark 2. The author's paper [17] has some inaccuracies in subsections 7.4,7.5, the author took into account the wrong sign for one of summands in the general solution. Really, the formula is more simple and the a priori estimates also.…”

Section: A Priori Estimatesmentioning

confidence: 99%

“…Such canonical domains can be a whole space R m , a half-space R m + = {x ∈ R m : x = (x , x m ), x m > 0} or a certain cone in R m . Some first author's results are included in the book [12], but to now the author have obtained some new results [13,14,16,17] which permit developing the approach more explicitly. Moreover, some results [19,21] can help to describe more complicated situations than ordinary m-dimensional cone in R m , namely we would like to consider here the situation when starting cone degenerates into a cone of a lower dimension.…”

Section: Introductionmentioning

confidence: 99%

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On Certain 3-Dimensional Limit Boundary Value Problems

Vasilyev

2020

Lobachevskii J Math

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The paper is devoted to studying limit behavior for a solution of model elliptic pseudodifferential equation with some integral boundary condition in 4-wedge conical canonical 3D singular domain with two parameters. It is shown that the solution of such boundary value problem can have a limit with respect to endpoint values of the parameters in appropriate Sobolev-Slobodetskii space if the boundary function is a solution of a special functional singular integral equation.

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“…We will use our constructions from [16,17] to describe the general solution of the equation ( 1) for the case ae…”

Section: A General Solutionmentioning

confidence: 99%

“…For first three cases 1, 2, 3 of Theorem 3 we can give an a priori estimate for the solution using the following lifting lemma; it was proved in [17].…”

Section: A Priori Estimatesmentioning

confidence: 99%

Section: A Priori Estimatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Certain 3-Dimensional Limit Boundary Value Problems

Vasilyev

2020

Lobachevskii J Math

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“…where B m is the operator generated by the Bochner kernel for the cone C and lf is an arbitrary extension of f on H s−α (R m ) [7,5]. The difficulty is that formula (4.2) yields a solution in a wider space, and there is arbitrariness in the choice of a solution in the required class H s (C).…”

Section: Coboundary Operators and Solvability Conditionsmentioning

confidence: 99%

Distributions Supported on Conical Surfaces and Generated Convolutions

Vasil’ev

2020

J Math Sci

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We describe the structure of distributions supported on conical surfaces and calculate the Fourier transform for some cones. The results are represented as convolutions with particular kernels. We use transmutation operators owing to which it is possible to clarify connections between the change of variables for a distribution and its Fourier transform. Bibliography: 17 titles.A lot of examples of distributions supported on various type surfaces in m-dimensional spaces can be found in [1,2]. However, in the literature, there are no results concerning distributions in the general form (counterparts of the Schwartz theorem on the general form of a distribution supported at a point in the one-dimensional case [3]). This paper is motivated by the recent results of [4]-[8], where pseudodifferential equations are studied in domains with conical points on the boundary in the multidimensional (m 3) case. Distributions and Change of Variables1.1. Choice of test functions. Let C be an acute convex cone in R m containing no entire line. Assume that the conical surface is given by the equation x m = ϕ(x ), x = (x 1 , . . . , x m−1 ), where ϕ : R m−1 → R is a smooth function on R m−1 \ {0} such that ϕ(0) = 0. We introduce the change of variables tand denote by T ϕ : R m → R m the change operator. It is obvious that this transformation is smooth except for the origin. We introduce the change of variables for the following class of distributions. For the space of test functions we take the Lizorkin space Φ(R m ) [9] which is a subspace of the Schwartz space S(R m ) of infinitely differentiable functions that are rapidly decreasing at infinity and vanish at the origin, together with all its derivatives. If Φ (R m ) and S (R m ) denote the corresponding spaces of distributions, then Φ (R m ) ⊃ S (R m ) and all operations with distributions in Φ (R m ) are legitimate for distributions in S (R m ).

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Distributions, Non-smooth Manifolds, Transmutations and Boundary Value Problems

Vasilyev

2020

Trends in Mathematics

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Pseudodifferential equations, wave factorization, and related problems

Cited by 20 publications

References 14 publications

On Certain 3-Dimensional Limit Boundary Value Problems

On Certain 3-Dimensional Limit Boundary Value Problems

Distributions Supported on Conical Surfaces and Generated Convolutions

Distributions, Non-smooth Manifolds, Transmutations and Boundary Value Problems

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