Dirichlet regularity of subanalytic domains
Tobias Kaiser · Transactions of the American Mathematical Society · 2008
Let ${\Omega }$ be a bounded and subanalytic domain in ${{\mathbb R}^n}$, ${n \geq 2}$. We show that the set of boundary points of ${\Omega }$ which are regular with respect to the Dirichlet problem is again subanalytic. Moreover, we give sharp upper bounds for the dimension of the set of irregular boundary points. This enables us to decide whether the domain has a classical Green function. In dimensions 2 and 3, this is the case, given some mild and necessary conditions on the topology of the domain.