Integral representations of a class of harmonic functions in the half space

Zhang, Yan Hui; Deng, Guan Tie; Qian, Tao

doi:10.1016/j.jde.2015.09.032

Cited by 7 publications

(4 citation statements)

References 3 publications

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“…In particular, this shows the strong influence that compactly supported data has on the growth or decay of solutions at infinity. The fact that our results are presented for compactly supported functions is not only for a better presentation, but also because general data yield a more complex problem and even for the Laplacian s = 1 this is an active research topic, see for example [28]. To mention one difficulty, without compact support bounds such as (1.7), (1.11), or (1.16) may not hold, and our uniqueness argument (see Lemma 3.4 below) cannot be applied; in fact, without growth assumptions uniqueness does not hold, by Proposition 1.4.…”

Section: Introductionmentioning

confidence: 99%

Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions

Abatangelo

Dipierro²,

Fall³

et al. 2019

Discrete &Amp; Continuous Dynamical Systems - A

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We present explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable extension of Dirichlet-type boundary conditions. A key ingredient in our proofs is a point inversion transformation which preserves harmonicity and allows us to use known results for the ball. We include uniqueness statements, regularity estimates, and describe the growth or decay of solutions at infinity and at the boundary.

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Section: Introductionmentioning

confidence: 99%

Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions

Abatangelo

Dipierro²,

Fall³

et al. 2019

Discrete &Amp; Continuous Dynamical Systems - A

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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%

“…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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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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“…In the present article, motivated by Zhang et al,() we study the Phragemén‐Lindelöf theorem for harmonic functions, and extend the Nevanlinna class for subharmonic functions from half complex plane to the half space of

R^{n}

. The approaches are nontrivial and the ideas are available in classical analysis as well.…”

Section: Introductionmentioning

confidence: 99%

The Phragemén‐Lindelöf principle of harmonic functions and conditions for nevanlinna class in the half space

Zhang

2019

Math Methods in App Sciences

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In this article, we firstly discuss a kind of Phragemén‐Lindelöf Principle of harmonic functions by using the Nevanlinna's representation in the high dimensional space. Secondly, we derive the sufficient conditions for subharmonic functions being in the Nevanlinna class as well. These results are main tools to study the Hilbert space of harmonic functions with analytic approaches and generalizations of some classic results.

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Integral representations of a class of harmonic functions in the half space

Cited by 7 publications

References 3 publications

Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions

Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions

Harmonic Functions in Slabs and Half-Spaces

The Phragemén‐Lindelöf principle of harmonic functions and conditions for nevanlinna class in the half space

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