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 $B\cap V=\emptyset$?

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    $\begingroup$ The answer is no if $n=2$. Indeed, every Jordan domain in the plane is regular for Dirichlet problem. $\endgroup$ Feb 18, 2020 at 12:22

1 Answer 1


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.


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