Let $U$ be an open subset of $\mathbb{R}^2$. Find the fundamental solution for the boundary value problems (BVP) \begin{align} \begin{cases} \Delta u+\lambda_i u=0&\text{in }U, &i=1,2.\\ \quad\qquad u=0 &\text{on }\partial U \end{cases} \end{align} where $\lambda_1>0$ and $\lambda_2<0$ are constants.
I want to find the Green’s functions for the above BVPs taking $U$ to be the unit ball. For that, I would need the fundamental solutions. Could you please give me some hints on how to find the fundamental solutions? Is there some better way to find the Green’s functions for these BVPs?
My comments:
- If $\lambda <0$, then by the weak maximum principle (Evans, PDE, Chapter 6) we get $u\equiv 0$.
- Assuming separation of variables $u(x,y)=X(x)Y(y)$ we arrive at the ODEs \begin{align} X’’-\mu X=0\\ Y’’+(\lambda+\mu)Y=0 \end{align} where $\mu$ is a constant whose sign would determine the solutions of the above ODEs. If we know the Green’s functions for the above ODEs, can we find the Green’s functions for the first BVPs in terms of these Green’s functions?
Thank you in advance.