Find a solution of the Laplace equation $-\Delta u=1$ with boundary condition $u=0$ on a spherical shell

Question

Let

• $n\ge 2$
• $B_\varepsilon$ and $\overline{B}_\varepsilon$ be the open and closed ball around $0$ with radius $\varepsilon>0$ in $\mathbb{R}^n$, respectively
• $R>0$, $\rho\in (0,R)$ and $\Omega:=B_R\setminus\overline{B}_\rho$

I'm searching for a solution of $$\left\{\begin{matrix}-\Delta u&=&1&\text{on }\Omega\\u&=&0&\text{on }\partial\Omega\end{matrix}\right.\tag{1}$$

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I know the solution of $(1)$ for $\rho=0$, i.e. $\Omega=B_R$. In that situation $$u(r)=\frac{R^2-r^2}{2n}$$ is a radial solution of $(1)$.

However, I'm unsure how I need to deal with the boundary condition and find a solution in this scenario.

Answer 1

If you deal with the spherical shell, you are still in the radial case. Hence, in the radial situation your problem is equivalent to $$ -u_{rr}-\frac{n-1}{r} u_r = 1, \\ u(\rho) = u(R) = 0, $$ where $r = |x|$.

The general solution is the following (due to WolframAlpha):

1) If $n=2$, then $$ u(r) = c_1 \ln r + c_2 - \frac{r^2}{4}. $$ 2) If $n \geq 3$, then $$ u(r) = \frac{1}{n} \left( \frac{c_1 n x^{2-n}}{2-n} - \frac{x^2}{2} \right) + c_2. $$

Using now the boundary data $u(\rho) = u(R) = 0$ you can uniquely define $c_1$ and $c_2$.