Is there an explicit expression of the Newtonian potential for torus? As the expression for ball is clear by calculation.
Definition of Newtonian potential of domain $\Omega$ at x is defined to be $u(x)=\int_{R^{n}}\Gamma(|x-y|)\chi_{\Omega}(y)dy$. When $x\in \Omega$ we say it is the interior Newtonian potential, otherwise we call it exterior Newtonian potential. Here $\Gamma(x)=\frac{1}{|x|^{n-2}}$ is the fundamental solution of Laplace, $\chi_{\Omega}$ is the characterization fucntion of $\Omega$