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.

  • 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
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    $\begingroup$ Yes (for example). $\endgroup$ – Igor Khavkine May 1 '15 at 13:47

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 $0$.


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