Integral formula for a Poisson equation

Question

The domain I am considering is $\mathbb{R}^3$ minus a sphere centered at the origin. So it is an unbounded domain.

I would like to know if there exists an integral formula for representing solutions $u$ of the Poisson equation $$-\Delta u=f,$$ with Neumann boundary conditions on the sphere $$\frac{\partial u}{\partial n}=g,$$ and another appropriate condition for the "boundary" at infinity, like $\|\nabla u\|=0$ at infinity.

In other words, I would like to know if there exists a similar formula to the Green's third identity for this setting.

Any reference would be greatly appreciated!

Thank you!

Answer 1

The Kelvin transform $$u^{*}(x^{*})= {\frac{1}{|x^{*}|^{{n-2}}}}u\left({\frac {R^{2}}{|x^{*}|^{2}}}x^{*}\right),\qquad x^{*}={\frac {R^{2}}{|x|^{2}}}x.$$ reduces the problem in exterior to a problem inside the ball, because the transformed function $u^*$ satisfies $$- \Delta u^{*}(x^{*})={\frac {R^{{4}}}{|x^{*}|^{{n+2}}}}f\left({\frac {R^{2}}{|x^{*}|^{2}}}x^{*}\right)$$ This also suggests an appropriate condition to impose at infinity: it is needed so that $u^*$ does not become singular at $0$.

The inhomogeneous exterior Neumann boundary condition becomes an inhomogeneous interior Neumann boundary condition, for a function $g^*$ that is related to $g$.

It remains to use the integral representation of solutions in a ball, based on Neumann Green's function. Luckily, the explicit form of this function is known for the case $n=3$, the source being the book Partial Differential Equations by Emmanuele DiBenedetto.