3
$\begingroup$

I am interesting in solutions to Poisson equation

$$\triangle \varphi = 4 \pi \rho \qquad (1)$$

defined on 3-dimensional oriented Riemannian manifold $(M,g)$, where $g$ is metric and $\triangle = \triangle[g]$ is Laplacian, and $\rho$ (density) is a smooth function on $M$ or a distribution. The first simple question: i) is it correct that for compact $M$ equation (1) has a solution if and only if $\int_{M} \rho dVol_g = 0$. If so, the fundamental solution to eq. (1) with $\rho = m \delta_p$, i.e.

$$\triangle \varphi = 4 \pi m \delta_p \qquad (2) $$

where $\delta_p$ is $\delta$-function located at point $p$ and $m \neq 0$ (mass), does not exist. However, in the case when $\rho = m_1 \delta_{p_1} + m_2 \delta_{p_2}$, where $p_1 \neq p_2$ and $m_1 + m_2 =0$ the solution to eq. (1) does exist - is it correct? Another two questions are about the solutions to eq. (2) for the case when : ii) $M = {\mathbb R} \times {\mathbb R} \times S^1$ and ii) $M = {\mathbb R} \times S^1 \times S^1$ with standards metrics (induced from ${\mathbb R}^3$). Could you give me some references, where explicit (analitic) fundamental solutions to Poisson eq. (2) with certain asymptotical conditions at infinity are written for these two cases.

$\endgroup$
  • 1
    $\begingroup$ If $M$ is non-compact, the integral over $M$ may not converge. Instead, from the equality $\triangle \varphi dVol_g = d*_g d\varphi$, you need the condition $[\rho dVol_g] = 0$ in de Rham cohomology $H^{\dim M}(M)$. In particular, $[\delta_p dVol_g] = 0$ for $M$ non-compact. For the second question, you may want to look up the method of images. $\endgroup$ – Igor Khavkine May 1 '15 at 8:18
  • $\begingroup$ It should be an infinite number of images (in both cases) which form a lattice. $\endgroup$ – Vladimir May 1 '15 at 11:53
  • 1
    $\begingroup$ Yes (for example). $\endgroup$ – Igor Khavkine May 1 '15 at 13:47
1
$\begingroup$

For the first question, in the compact case, one can define a fundamental solution $G(x, y)$ by

$ \Delta G(x, y) = \delta_y - \frac{1}{vol(M)} $,

so that the right-hand side has zero average. Such a function $G(x,y)$ exists for any compact Riemannian manifold (see Thierry Aubin, Some Nonlinear Problems in Riemannian Geometry), and it can be used to solve Poisson's equation with an arbitrary right-hand side in the same way as the conventional fundamental solution.

Combining these functions $G(x,y)$ for different $y$'s, you can, in particular, solve Poisson's equation whose right hand-side is a combination of delta functions, provided that the coefficients add up to $0$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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