Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How to prove that a function

$$v_H(\mathbf{r}) = \int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}d\mathbf{r'}$$

is finite at any $\textbf{r}$?

$\rho(\mathbf{r'})$ appearing inside the integral is some well-behaved function that is finite everywhere, decays exponentially and vanishes at infinity.

Would a function

$$v(\mathbf{r}) = \int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|^n}d\mathbf{r'}$$

(where $n$ is some nonnegative integer)

be also finite at any $\mathbf{r}$?

share|improve this question
    
Do we assume $\textbf{r}\in\mathbb{R}^3$? –  icurays1 Nov 21 '12 at 17:11
    
yes, we are considering 3d space. –  molkee Nov 21 '12 at 21:42
add comment

1 Answer 1

up vote 1 down vote accepted

Partial answer: If $\rho(\textbf{r}^\prime)\in C^2_c(\mathbb{R}^3)$ (twice continuously differentiable with compact support), then $v_H(\textbf{r})\in C^2(\mathbb{R}^3)$ and $v_H$ solves a Poisson equation:

$$ \triangle v_H=4\pi\rho $$

This is Theorem 1, page 23-25 in L. Evans' book on Partial Differential Equations, paraphrased in some lecture notes here. Note of course that care must be taken near $\textbf{r}=\textbf{r}^\prime$ (this is addressed in the proof).

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.