I am thinking of the Dirichlet boundary condition $u|_{\partial \Omega}$ for a domain $\Omega \subset \mathbb{R}^n$. Let $\Delta$ be the Dirichlet Laplacian, which accepts only functions with the above Dirichlet boundary condition (http://en.wikipedia.org/wiki/Dirichlet_eigenvalue). Now let $\Phi(x,y)$ be the Green's function of $\Delta-id$. What is the local behaviour of $\Phi(x,y)$? Is it like $$ \Phi(x,y)\sim \frac{1}{|x-y|^{n-2}} \ (n >2), \ \ \ \Phi(x,y)\sim \log|x-y| \ (n=2) $$ just like free boundary case? Or we have to consider something else?


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