# The converse of a Poincaré's result on regular boundary points

Let $$V$$ be a bounded open set in $$\mathbb{R}^n$$ with $$n>1$$. According to a well known result due to Poincaré, if $$x$$ is a point in the boundary $$\partial V$$ and there exists a ball $$B$$ such that $$x\in\partial B$$ and $$B\cap V=\emptyset$$, then $$x$$ is a regular point for the Dirichlet problem.

Is there a converse for this result? More generally, under which condition(s) can we say that if $$x\in\partial V$$, there exists a ball $$B$$ such that $$x\in\partial B$$ and $$B\cap V=\emptyset$$?

• The answer is no if $n=2$. Indeed, every Jordan domain in the plane is regular for Dirichlet problem. Feb 18 '20 at 12:22

The answer is no. Take the unit ball in $$R^3$$, and remove from it the halfplane $$x_3=0, x_1\geq 0$$. This is regular.