Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Integral Representations of solutions to Poisson Equations

Aye Chan May, Adisak Seesanea

TL;DR

The paper develops a constructive framework for integral representations of Poisson equation solutions in $\mathbb{R}^n$ by convolving the data $f$ with the Newtonian fundamental solution $E_n$. It proves well-definedness, continuity, and higher-order differentiability of the potential $u$ under precise integrability (and in some cases differentiability) conditions on $f$, establishing explicit derivative formulas such as $\partial_{i}u(x)=\int_{\mathbb{R}^n}\partial_{i}E_n(x-y)f(y)\,dy$ and $\partial_{i}\partial_{j}u(x)=\int_{\mathbb{R}^n}\partial_{j}E_n(x-y)\partial_{i}f(y)\,dy$, with $\Delta u=f$. It further extends these results to Lorentz-space data to obtain existence and uniqueness of bounded solutions to the Poisson problem, and discusses generalizations to positive homogeneous kernels. These results provide robust, constructive criteria for when Newtonian potentials yield regular solutions to Poisson equations, including unbounded domains and low-regularity data.

Abstract

We give a constructive approach for the study of integral representations of classical solutions to Poisson equations under some integrability conditions on data functions.

Integral Representations of solutions to Poisson Equations

TL;DR

The paper develops a constructive framework for integral representations of Poisson equation solutions in by convolving the data with the Newtonian fundamental solution . It proves well-definedness, continuity, and higher-order differentiability of the potential under precise integrability (and in some cases differentiability) conditions on , establishing explicit derivative formulas such as and , with . It further extends these results to Lorentz-space data to obtain existence and uniqueness of bounded solutions to the Poisson problem, and discusses generalizations to positive homogeneous kernels. These results provide robust, constructive criteria for when Newtonian potentials yield regular solutions to Poisson equations, including unbounded domains and low-regularity data.

Abstract

We give a constructive approach for the study of integral representations of classical solutions to Poisson equations under some integrability conditions on data functions.
Paper Structure (3 sections, 7 theorems, 137 equations)

This paper contains 3 sections, 7 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fundamental Solutions

Key Result

Theorem 1.1

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a Lebesgue measurable function. Define If $\frac{f(y)}{1+|y|^{n-2}} \in L^1(\mathbb{R}^n)$, $\frac{\nabla f(y)}{1+|y|^{n-1}} \in L^1(\mathbb{R}^n)$ and $f \in \mathcal{C}^{1}(\mathbb{R}^n)$, then $u$ is well-defined in $\mathbb{R}^n$ and belongs to $\mathcal{C}^{2}(\mathbb{R}^n)$. Moreover, for each $i,j \in \lbrace 1,...,n\rbrace$ and $ Furthermore

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 2.1: Mean-value formulas for Laplace eqaution
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.1