The punctured disc has an irregular boundary point and a continuous boundary datum with no harmonic solution

Suppose $u$ were a continuous harmonic solution of this boundary-value problem on $\mathrm{\Omega }$. Then $u$ is bounded on every punctured neighbourhood of $0$ because it extends continuously to the puncture with value $1$. By [L1], $u$ extends to a harmonic function $U$ on the full unit disc.

We determine the Perron envelope directly. The constant $0$ is a Perron lower function for $\phi$, so $U_{\phi }\ge 0$. Let $v$ be any other lower function and fix $\epsilon >0$. Its boundary limsup at the puncture gives $v\le 1+\epsilon$ on all sufficiently small circles about $0$. The boundary limsup at each point of the compact unit circle gives, by a finite subcover, some $R<1$ such that $v\le \epsilon$ whenever $R\le \mathrm{\mid }z\mathrm{\mid }<1$. For any $0<\delta <R$ sufficiently small, compare $v$ on $\delta <\mathrm{\mid }z\mathrm{\mid }<R$ with the harmonic function [L3, L5, given] $h_{\delta ,R}(z)=\epsilon +\log (R\mathrm{/}\mathrm{\mid }z\mathrm{\mid })\mathrm{/}\log (R\mathrm{/}\delta )$. The inner and outer boundary bounds give $v-h_{\delta ,R}\le 0$ on both circles; [L5] and the maximum principle therefore give $v\le h_{\delta ,R}$ throughout the annulus. For a fixed $z$ with $\mathrm{\mid }z\mathrm{\mid }<R$, letting $\delta \downarrow 0$ yields $v(z)\le \epsilon$. For any fixed $z\in \mathrm{\Omega }$, take $R$ above $\mathrm{\mid }z\mathrm{\mid }$ and sufficiently close to $1$, then let $\epsilon \downarrow 0$. Thus every lower function is at most $0$, so $U_{\phi }=H_{\phi }=0$.

The extension $U$ still has boundary value $0$ on the unit circle, so [L2] forces $U\equiv 0$ on the closed unit disc. But then $U(0)=0$, contradicting the prescribed puncture value $1$. Therefore no such harmonic solution exists.

In particular, $H_{\phi }(z)\to 0$ as $z\to 0$ inside $\mathrm{\Omega }$, whereas $\phi (0)=1$. By [L4] the puncture is irregular. Together with step 2.1, this proves both asserted failures for the given domain and datum.