Zero extension for Poisson’s equation

In this paper, we present a necessary and sufficient condition to guarantee that the extended function of the solution for Poisson's equation in a smaller domain by zero extension is still the solution of the corresponding extension problem in a larger domain. We prove the results under the frameworks of classical solutions, strong solutions and weak solutions. Furthermore, we give some observations for the nonlinear pLaplace equation.     

Heinz type inequalities for mappings satisfying Poisson’s equation

In this paper, we study the Heinz type inequalities for mappings satisfying Poisson’s equation. Some results generalize the ones obtained by Partyka and Sakan.     

Cauchy transform and Poisson’s equation

Let $u\in W^{1,p}\cap W_0^{1,p}$, $1?p?\infty$ be a solution of the Poisson equation $\Delta u=h$, $h\in L^p$, in the unit disk. We prove ${6?u6}_{L^p}?a_p{6h6}_{L^p}$ and ${6?u6}_{L^p}?b_p{6h6}_{L^p}$ with sharp constants $a_p$ and $b_p$, for $p=1$, $p=2$, and $p=\infty$. In addition, for $p>2$, with sharp constants $c_p$ and $C_p$, we show ${6?u6}_{L^{\infty }}?c_p{6h6}_{L^p}$ and ${6?u6}_{L^{\infty }}?C_p{6h6}_{L^p}$. We also give an extension to smooth Jordan domains.These problems are equivalent to determining a precise value of the $L^p$ norm of the Cauchy transform of Dirichlet’s problem.     

Coefficient inverse problem for Poisson’s equation in a cylinder

The inverse problem of determining the coefficient on the right-hand side of Poisson’s equation in a cylindrical domain is considered. The Dirichlet boundary value problem is studied. Two types of additional information (overdetermination) can be specified: (i) the trace of the solution to the boundary value problem on a manifold of lower dimension inside the domain and (ii) the normal derivative on a portion of the boundary. (Global) existence and uniqueness theorems are proved for the problems. The study is performed in the class of continuous functions whose derivatives satisfy a Hölder condition.     

Construction of polynomial solutions to some boundary value problems for Poisson’s equation

A polynomial solution of the inhomogeneous Dirichlet problem for Poisson’s equation with a polynomial right-hand side is found. An explicit representation of the harmonic functions in the Almansi formula is used. The solvability of a generalized third boundary value problem for Poisson’s equation is studied in the case when the value of a polynomial in normal derivatives is given on the boundary. A polynomial solution of the third boundary value problem for Poisson’s equation with polynomial data is found.     

A boundary value problem for Poisson’s two-media equation inL p /2 -spaces

In the paper we study a binding boundary value problem for two media for Poisson's equation μΔu=f(x) with solutions in the class , 1<p<∞, with the corresponding seminorm, where

On Lipschitz continuity of solutions of hyperbolic Poisson’s equation

In this paper, we investigate solutions of the hyperbolic Poisson equation \(\Delta _{h}u(x)=\psi (x)\), where \(\psi \in L^{\infty }(\mathbb {B}^{n}, {\mathbb R}^n)\) and

$$\begin{aligned} \Delta _{h}u(x)= (1-|x|^2)^2\Delta u(x)+2(n-2)\left( 1-|x|^2\right) \sum _{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \end{aligned}$$

is the hyperbolic Laplace operator in the n-dimensional space \(\mathbb {R}^n\) for \(n\ge 2\). We show that if \(n\ge 3\) and \(u\in C^{2}(\mathbb {B}^{n},{\mathbb R}^n) \cap C(\overline{\mathbb {B}^{n}},{\mathbb R}^n )\) is a solution to the hyperbolic Poisson equation, then it has the representation \(u=P_{h}[\phi ]-G_{ h}[\psi ]\) provided that \(u\mid _{\mathbb {S}^{n-1}}=\phi \) and \(\int _{\mathbb {B}^{n}}(1-|x|^{2})^{n-1} |\psi (x)|\,d\tau (x)<\infty \). Here \(P_{h}\) and \(G_{h}\) denote Poisson and Green integrals with respect to \(\Delta _{h}\), respectively. Furthermore, we prove that functions of the form \(u=P_{h}[\phi ]-G_{h}[\psi ]\) are Lipschitz continuous.

     

Regularity of solutions of Poisson’s equation in multiplier spaces

The most important result stated in this paper is to show that the solutions of the Poisson equation −Δu = f, where f ∈ (Ḣ¹(ℝ ^d ) → (Ḣ⁻¹(ℝ ^d )) is a complex-valued distribution on ℝ ^d , satisfy the regularity property D ^k u ∈ (Ḣ¹ → Ḣ⁻¹) for all k, |k| = 2. The regularity of this equation is well studied by Maz’ya and Verbitsky [12] in the case where f belongs to the class of positive Borel measures.      

Convergence of the method of fundamental solutions for Poisson’s equation on the unit sphere

Solutions of boundary value problems of the Laplace equation on the unit sphere are constructed by using the fundamental solution

Grid independent convergence using multilevel circulant preconditioning: Poisson’s equation

BIT Numerical Mathematics - We consider the iterative solution of the discrete Poisson’s equation with Dirichlet boundary conditions. The discrete domain is embedded into an extended domain...     

Transport equation on semidiscrete domains and Poisson–Bernoulli processes

In this paper, we consider a scalar transport equation with constant coefficients on domains with discrete space and continuous, discrete or general time. We show that on all these underlying domains, solutions of the transport equation can conserve sign and integrals both in time and space. Detailed analysis reveals that, under some initial conditions, the solutions correspond to counting stochastic processes and related probability distributions. Consequently, the transport equation could generate various modifications of these processes and distributions and provide some insights into corresponding convergence questions. Possible applications are suggested and discussed.     

Discretization of solutions to Poisson’s equation in the Korobov class

A sharp discretization error estimate on the power scale is obtained for the solution of Poisson’s equation with a right-hand side from the Korobov class with the application of Smolyak grid nodes. In some cases, the results coincide in the order of the upper error estimate with earlier results of other authors, but the discretization operator proposed is simpler.     

Functional a posteriori estimates for Maxwell’s equation

The note is concerned with functional type a posteriori estimates of the difference between exact and approximate solutions of the Maxwell problem . The estimates are derived by transformations of the basic integral identity defining a generalized solution to the problem by using the method suggested by the author. The estimates are obtained in the case ℵ > 0 and ℵ = 0. Bibliography: 10 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 83–88.     

Van Vleck’s functional equation for the sine

We solve Van Vleck’s functional equation on semigroups with an involution in terms of multiplicative functions.     

A hybrid method for numerical solution of Poisson’s equation in a domain with a dielectric corner

An electrostatic problem of determining a potential in a domain containing an incoming dielectric corner, which reduces to solving Poisson’s equation in this domain, is considered. A specific feature of the solution of this problem is that it is bounded in a neighborhood of the dielectric corner but its gradient increases without limit. An efficient hybrid algorithm for the numerical solution of the problem, based on the finite element method and taking into account the known asymptotic representation of the solution in the neighborhood of the dielectric corner, is proposed.     

On convergence of the method of fundamental solutions for solving the Dirichlet problem of Poisson’s equation

In this paper the convergence of using the method of fundamental solutions for solving the boundary value problem of Laplaces equation in R² is established, where the boundaries of the domain and fictitious domain are assumed to be concentric circles. Fourier series is then used to find the particular solutions of Poissons equation, which the derivatives of particular solutions are estimated under the L² norm. The convergent order of solving the Dirichlet problem of Poissons equation by the method of particular solution and method of fundamental solution is derived. Dedicated to Charles A. Micchelli with esteem on the occasion of his 60th birthdayAMS subject classification 35J05, 31A99