On the Poisson Representation of a Function Harmonic in the Upper Half-Plane

Gergün, Seçil; Островскии, И. В.

doi:10.1007/bf03321016

Cited by 3 publications

(5 citation statements)

References 7 publications

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“…Modifications to the classical Poisson kernel (38) that can be used with continuous functions f (x) that enjoy no greater than polynomial growth as |x| tends to ∞ can be found in [3,4,14,15,17,24,32] where the rate of growth of f determines the necessary modifications. Such kernels can also be used with more general initial data f ; for example, various classes of measures and locally integrable functions are treated in most of the abovementioned works.…”

Section: Basic Formulasmentioning

confidence: 99%

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Harmonic Functions in Slabs and Half-Spaces

Madych

2020

Springer Optimization and Its Applications

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The usual solution to the Dirichlet problem for the Laplace equation Δu = 0 in the slab R n × (a, b), where −∞ < a < b < ∞, and the half-space R n ×(0, ∞) involves convolution of the data with a Poisson kernel. Interestingly, the class of distributions which is convolvable with the natural Poisson kernel Q for the slab is considerably wider than that which is convolvable with the classical Poisson kernel P for the half-space. We investigate this curious phenomenon and observe that arbitrary tempered distributions can be convolved with Q, resulting in functions harmonic in the slab with no greater than polynomial growth in the interior and distributionally bounded on hyperplanes parallel to the boundary. Conversely, we show that all harmonic functions in the slab which enjoy no greater than polynomial growth in the interior and are distributionally bounded on hyperplanes parallel to the boundary can be characterized as Poisson integrals of tempered distributions. In the case of the half-space we observe that the classical Poisson kernel can be modified so that the result is applicable to all tempered distributions and gives rise to harmonic functions in the half-space with the prescribed boundary values. In both cases if the boundary data is given by polynomials then so is the resulting harmonic function. In the appendix we record some additional properties of the kernel Q and offer several pertinent comments and observations.

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Section: Basic Formulasmentioning

confidence: 99%

“…This issue can be dealt with in certain cases by modifying the classical Poisson kernel P appropriately; examples can be found in [3,4,14,15,17,24,32] and elsewhere. Nevertheless the difference in behavior of the kernels P and Q is a curious phenomenon which deserves further study.…”

Section: Introductionmentioning

confidence: 99%

Harmonic Functions in Slabs and Half-Spaces

Madych

2020

Springer Optimization and Its Applications

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“…Here, c 2 = 2 and c n = (n − 2)w n when n ≥ 3. This definitions appears in many different contexts, we refer the reader to [3,4,10,15,20] and references therein. It follows from our assumption on Γ that PI Γ (V, W ) is continuous on ∂H(Γ) (see [5,Theorem 6.15]).…”

Section: Preliminariesmentioning

confidence: 99%

“…respectively. Hence E (2) and E (3) have also two coverings {r (2) j } ∞ j=0 and {r By rearranging three coverings {r…”

Section: Applications To the Geometrical Property Of Rarefied Setmentioning

confidence: 99%

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On the cylindrical Green's function for representation theory and its applications

Qiao¹

2019

Ann. Acad. Sci. Fenn. Math.

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It has been known for some time that the Green's function of a planar domain can be defined in terms of the exit time of Brownian motion, and this definition has been applied to the representation theory. In this paper, we extend the notion of conformal invariance of Green's function to superharmonic functions, and use this extension to construct and estimate the cylindrical Green's function for Dirichlet boundary value problem. These considerations lead to a new proof of the representation theory of superharmonic functions defined in cylinders. We also show this estimation can be used to obtain a covering property of rarefied sets at infinity. Finally, by giving an example, we show that the reverse of this property is not true.

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Explicit solutions of the Dirichlet–Schrödinger problem via the new cylindrical Poisson–Schrödinger kernel method

Qiao

2023

Forum Mathematicum

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In this paper, we study the cylindrical Schrödinger operator and construct a modified cylindrical Poisson–Schrödinger kernel, in analogy to the classical theory of the Laplacian in a strip. A new method expressing explicitly the modified cylindrical Poisson–Schrödinger kernels of any degree is obtained, and a formula expressing the modified cylindrical Poisson–Schrödinger integrals in terms of Nevanlinna–Schrödinger norms is deduced. Applications in the solutions of Dirichlet–Schrödinger problem are described. More specifically, the existence and uniqueness of explicit solutions to the problem mentioned above for the cylindrical Schrödinger equation are obtained. It is shown that our method yields more explicit results than do other methods. Much more challenging geometric configurations, such as a cylinder with a rough base bear new and interesting challenges arising from the lateral boundary conditions. These have-at least to our knowledge-not been addressed before.

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On the Poisson Representation of a Function Harmonic in the Upper Half-Plane

Cited by 3 publications

References 7 publications

Harmonic Functions in Slabs and Half-Spaces

Harmonic Functions in Slabs and Half-Spaces

On the cylindrical Green's function for representation theory and its applications

Explicit solutions of the Dirichlet–Schrödinger problem via the new cylindrical Poisson–Schrödinger kernel method

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