scholarly journals The Dirichlet problem for standard weighted Laplacians in the upper half plane

2016 ◽  
Vol 436 (2) ◽  
pp. 868-889 ◽  
Author(s):  
Marcus Carlsson ◽  
Jens Wittsten
Keyword(s):  
Dirichlet Problem ◽  
Half Plane ◽  
Upper Half Plane

scholarly journals Scattering on periodic metric graphs

2020 ◽  
Vol 32 (08) ◽  
pp. 2050024
Author(s):  
Evgeny Korotyaev ◽  
Natalia Saburova
Keyword(s):  
Dirichlet Problem ◽  
Half Plane ◽  
Fredholm Determinant ◽  
Unit Interval ◽  
Discrete Laplacian ◽  
Metric Graphs ◽  
Wave Operators ◽  
S Matrix ◽  
Upper Half Plane ◽  
Uniformly Bounded

We consider the Laplacian on a periodic metric graph and obtain its decomposition into a direct fiber integral in terms of the corresponding discrete Laplacian. Eigenfunctions and eigenvalues of the fiber metric Laplacian are expressed explicitly in terms of eigenfunctions and eigenvalues of the corresponding fiber discrete Laplacian and eigenfunctions of the Dirichlet problem on the unit interval. We show that all these eigenfunctions are uniformly bounded. We apply these results to the periodic metric Laplacian perturbed by real integrable potentials. We prove the following: (a) the wave operators exist and are complete, (b) the standard Fredholm determinant is well-defined and is analytic in the upper half-plane without any modification for any dimension, (c) the determinant and the corresponding S-matrix satisfy the Birman–Krein identity.

A Dirichlet Problem for the Inhomogeneous Polyharmonic Equation in the Upper Half Plane

2007 ◽  
Vol 14 (1) ◽  
pp. 33-52
Author(s):  
Heinrich Begehr ◽  
Evgenija Gaertner
Keyword(s):  
Dirichlet Problem ◽  
Green Function ◽  
Differential Operator ◽  
Integral Representation ◽  
Half Plane ◽  
Representation Formula ◽  
Higher Order ◽  
Polyharmonic Equation ◽  
Integral Representation Formula ◽  
Upper Half Plane

Abstract On the basis of a higher order integral representation formula related to the polyharmonic differential operator and obtained through a certain polyharmonic Green function, a Dirichlet problem is explicitly solved in the upper half plane.

scholarly journals OVERVIEW OF VNC PUBLICATIONS 

2018 ◽  
Author(s):  
Г.П. Селиверстова
Keyword(s):  
Fourier Transform ◽  
Dirichlet Problem ◽  
Half Plane ◽  
Boundary Function ◽  
Optimal Recovery ◽  
Recovery Method ◽  
Finite Segment ◽  
Plane Parallel ◽  
Straight Line ◽  
Upper Half Plane

Работа посвящена задаче наилучшего восстановления реше- ния задачи Дирихле в пространстве квадратично суммируемых функций на прямой в верхней полуплоскости, параллельной оси абсцисс, по следующей информации о граничной функции: гра- ничная функция принадлежит некоторому соболевскому про- странству функций, а ее преобразование Фурье известно при- ближенное на конечном отрезке, симметричном относительно нуля. Построен оптимальный метод восстановления и найдено точное значение погрешности оптимального восстановления. The work is devoted to the problem of the best recovery solution- the Dirichlet problem in the space of quadratically summable functions on a straight line in the upper half plane parallel to the axis abscissa, on the following information on the boundary function: gra- personal function belongs to some Sobolev Pro- the wandering of functions, and its Fourier transform is known in- near on a finite segment, symmetric with respect to zero's. The optimal recovery method was constructed and found the exact error value of the optimal recovery.

scholarly journals On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

2014 ◽  
Vol 57 (2) ◽  
pp. 381-389
Author(s):  
Adrian Łydka
Keyword(s):  
Functional Equation ◽  
Meromorphic Function ◽  
Elliptic Curve ◽  
Explicit Formula ◽  
Half Plane ◽  
Complex Plane ◽  
Möbius Function ◽  
Mobius Function ◽  
Upper Half Plane ◽  
Explicit Formulae

AbstractWe study analytic properties function m(z, E), which is defined on the upper half-plane as an integral from the shifted L-function of an elliptic curve. We show that m(z, E) analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for m(z, E) in the strip |ℑz| < 2π.

Exact determination of gravity stresses in finite elastic slopes: Part I. Theoretical considerations

1983 ◽  
Vol 20 (1) ◽  
pp. 47-54 ◽  
Author(s):  
V. Silvestri ◽  
C. Tabib
Keyword(s):  
Plane Strain ◽  
Half Plane ◽  
Inclination Angle ◽  
Complex Variable ◽  
Earth Pressure ◽  
Exact Distributions ◽  
Exact Determination ◽  
Upper Half Plane ◽  
Theoretical Considerations

The exact distributions of gravity stresses are obtained within slopes of finite height inclined at various angles, −β (β = π/2, π/3, π/4, π/6, and π/8), to the horizontal. The solutions are obtained by application of the theory of a complex variable. In homogeneous, isotropic, and linearly elastic slopes under plane strain conditions, the gravity stresses are independent of Young's modulus and are a function of (a) the coordinates, (b) the height, (c) the inclination angle, (d) Poisson's ratio or the coefficient of earth pressure at rest, and (e) the volumetric weight. Conformal applications that transform the planes of the various slopes studied onto the upper half-plane are analytically obtained. These solutions are also represented graphically.

scholarly journals Revisiting the Siegel upper half plane II

2004 ◽  
Vol 376 ◽  
pp. 45-67 ◽  
Author(s):  
Pedro J. Freitas ◽  
Shmuel Friedland
Keyword(s):  
Half Plane ◽  
Upper Half Plane
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