# Sequence of harmonic measure

There is a well-known result stating that if $$\mu_{n}$$ is a sequence of uniformly bounded measures on a compact set $$E$$ of $$\mathbb{R}^{m}$$, then there is a subsequence $$\mu_{n_{j}}$$ that converges weakly to some measure $$\mu$$ on $$E$$. Now, suppose $$D$$ is a bounded domain of $$\mathbb{R}^{m}$$ with $$m>1$$ and $$y$$ a point on the boundary $$\partial D$$ of $$D$$. Take a sequence $$x_{n}$$ in $$D$$ that converges to $$y$$ and consider the sequence $$\omega(x_{n},D)$$ (harmonic measure of $$D$$ at $$x_{n}$$. This sequence has a subsequence that converges weakly to some $$\mu$$ on $$\partial D$$. My question is: is $$\mu$$ known? What do we know about it?

• Usually, this is the Dirac measure at $y$. (Unless $y$ is irregular for the Dirichlet problem.) – Mateusz Kwaśnicki Oct 30 '19 at 13:18
• Thanks. Now suppose that for some continuous function $f$ on $\partial D$, there is a subset $A$ of $\partial D$ that does NOT contain $y$ and $$\int_{\partial D}fd\omega(x_{n},D)=\int_{A}fd\omega(x_{n},D).$$ Can $y$ be steel a regular point and the subsequence converge to $f(y)$?! – M. Rahmat Oct 30 '19 at 14:16
• Sorry, I do not get your comment. Set $f = 0$ on $\partial D$ (or at least on $(\partial D) \setminus A$). Your condition is then obviously satisfied, but why would it imply anything about regularity of a boundary point $y$? – Mateusz Kwaśnicki Oct 30 '19 at 21:44
• Suppose for instance that $A$ is relatively close in $\partial D$. Then the above integral of the right equals $$\int_{\partial D}\chi_{A}fd\omega(x_{n},D)\to\chi_{A}(y)f(y)=0$$ for all continuous function $f$, because $y$ doesn't belong to $A$. – M. Rahmat Oct 30 '19 at 22:32
• Your last comment states the following: if $A$ is a closed subset of $\partial D$ and $y \in (\partial D) \setminus A$, then the solution of the Dirichlet problem with boundary data $\chi_A f$ converges to zero at $y$. This statement is equivalent to $y$ being regular for the Dirichlet problem in $D$. And what is the question? – Mateusz Kwaśnicki Oct 30 '19 at 22:50