On zeros, singular boundary functions, and modules of angular boundary values for one class of functions analytic in a half-plane

In this note we give a simple proof of an operator-interpolation theorem (Theorem 2) due originally to Donoghue [6], and Lions-Foias [7].Let be the complex plane, the open upper half-plane, the real line, ℛ+ and ℛ- the non-negative and non-positive axes. Denote by the class of positive functions on which extend analytically to —ℛ-, and map into itself. Denote by ’ the class of functions φ such that φ(x1/2)2 is in .

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On boundedness and angular boundary values of subharmonic functions in classes ℜθ

Russian Mathematics ◽

10.3103/s1066369x1904008x ◽

2019 ◽

Vol 63 (4) ◽

pp. 75-78

Author(s):

S. L. Berberyan

Keyword(s):

Boundary Values ◽

Subharmonic Functions ◽

Angular Boundary

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Polynomial Solutions of the Dirichlet Problem for the Tricomi Equation in a Strip

Mathematics and Mathematical Modeling ◽

10.24108/mathm.0318.0000120 ◽

2018 ◽

pp. 1-12

Author(s):

O. D. Algazin

Keyword(s):

Dirichlet Problem ◽

Half Plane ◽

Mixed Type ◽

Polynomial Growth ◽

Polynomial Solution ◽

Transform Method ◽

Polynomial Solutions ◽

Tricomi Equation ◽

Equation Of Mixed Type ◽

Class Of Functions

In the paper we consider the Tricomi equation of mixed type. This equation is elliptic in the upper half-plane, hyperbolic in the lower half-plane and parabolically degenerate on the boundary of half-planes. Equations of a mixed type are used in transonic gas dynamics. The Dirichlet problem for an equation of mixed type in a mixed domain is, in general, ill- posed. Many papers has been devoted to the search for conditions for the well-posednes of the Dirichlet problem for a mixed-type equation in a mixed domain.This paper is devoted to finding exact polynomial solutions of the inhomogeneous Tricomi equation in a strip with a polynomial right-hand side. The Fourier transform method shows that the Dirichlet boundary value problem with polynomial boundary conditions has a polynomial solution. An algorithm for constructing this polynomial solution is given and examples are considered. If the strip lies in the ellipticity region of the equation, then this solution is unique in the class of functions of polynomial growth. If the strip lies in a mixed domain, then the solution of the Dirichlet problem is not unique in the class of functions of polynomial growth, but it is unique in the class of polynomials.

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A Note on the Displacement Problem of Elastostatics with Singular Boundary Values

Axioms ◽

10.3390/axioms8020046 ◽

2019 ◽

Vol 8 (2) ◽

pp. 46 ◽

Cited By ~ 2

Author(s):

Alfonsina Tartaglione

Keyword(s):

Elastic Body ◽

Function Class ◽

Singular Boundary ◽

Regular Boundary ◽

Boundary Values ◽

Exterior Domains ◽

Displacement Problem ◽

Natural Function ◽

Linear Elastostatics ◽

Class C

The displacement problem of linear elastostatics in bounded and exterior domains with a non-regular boundary datum a is considered. Precisely, if the elastic body is represented by a domain of class C k ( k ≥ 2 ) of R 3 and a ∈ W 2 − k − 1 / q , q ( ∂ Ω ) , q ∈ ( 1 , + ∞ ) , then it is proved that there exists a solution which is of class C ∞ in the interior and takes the boundary value in a well-defined sense. Moreover, it is unique in a natural function class.

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Simulation of Seismic Wave Scattering by Embedded Cavities in an Elastic Half-Plane Using the Novel Singular Boundary Method

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.oa-2016-0187 ◽

2018 ◽

Vol 10 (2) ◽

pp. 322-342 ◽

Cited By ~ 123

Author(s):

Ji Lin

Keyword(s):

Half Plane ◽

Seismic Wave ◽

Wave Scattering ◽

Singular Boundary ◽

The Novel ◽

Boundary Method ◽

Seismic Wave Scattering ◽

Singular Boundary Method

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Nonsteady frictional heating on a sliding contact

MATEC Web of Conferences ◽

10.1051/matecconf/201822603030 ◽

2018 ◽

Vol 226 ◽

pp. 03030

Author(s):

Vladimir B. Zelentsov ◽

Boris I. Mitrin

Keyword(s):

Integral Equation ◽

Singular Integral Equation ◽

Contact Problem ◽

Half Plane ◽

Frictional Heating ◽

Contact Stresses ◽

Sliding Contact ◽

Singular Equation ◽

Static Contact ◽

Class Of Functions

We consider quasi-static contact problem on frictional heating on a sliding contact of a rotating rigid cylinder and a half-plane. The cylinder is pressed towards the half-plane material. The problem is reduced to solution of a singular integral equation with respect to contact stresses. Solution of the singular equation is looked for in a class of functions limited on the edge, with two additional conditions to determine timedependent boundaries of the contact area. Temperature at the contact and inside the half-plane is determined in terms of contact stresses.

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