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An open set satisfies an exterior cone condition at $z$ if a solid cone with vertex $z$ lies outside the set in some neighbourhood of $z$. The stronger global version requires the closures of the cone and the set to meet only at $z$. In two dimensions, a power barrier for an exterior cone makes such a point a regular boundary point for the Dirichlet problem. Local cone conditions suffice for local boundary regularity; a global cone is an additional geometric assumption.

For a bounded domain whose boundary points are all regular boundary points, continuous Dirichlet boundary data $f$ have a unique solutionHere ${\tau }_D$ is the Brownian exit time. The Strong Markov property gives the mean value property for harmonic functions; a barrier for the Dirichlet problem gives continuity at the boundary. This is an existence theorem as well as the Brownian representation of the Dirichlet problem for an already known solution.

If $B(y,R)$ is an exterior ball tangent to a planar domain at $x_0$, thenis a harmonic function in the domain, vanishes at $x_0$, and is positive at every other point of its closure. It is therefore a barrier for the Dirichlet problem and proves that $x_0$ is a regular boundary point.

A standard form of the Kakutani solution of the Dirichlet problem uses a bounded domain $D\subset {\mathbb{R}}^d$, continuous Dirichlet boundary data $f\in C(\partial D)$, and regularity of every boundary point for the Dirichlet problem. The boundedness of $D$ makes $\partial D$ a compact set, so $f$ is a bounded function and a uniformly continuous function. A bounded domain with $C^2$ boundary is a sufficient geometric case; one must not omit boundary regularity for an arbitrary bounded domain.

One standard analytic characterization of a regular boundary point on a bounded domain is the existence of a positive harmonic barrier: for each $\xi \in \partial D$ there is a harmonic function $h_{\xi }\in C(\overline{D})$ with $h_{\xi }(\xi )=0$ and $h_{\xi }(x)>0$ for $x\in \overline{D}\setminus \{\xi \}$. This is the harmonic form of a barrier for the Dirichlet problem; the barrier characterization of regularity is a standard fact of potential theory.

Let $B$ be $d$-dimensional Brownian motion started at $x\in D$ and let ${\tau }_D=\inf \{t\ge 0:B_t\notin D\}$ be its Brownian exit time. Then the unique solution in $C(\overline{D})\cap C^2(D)$ isEquivalently, $u(x)={\int }_{\partial D}f(\xi ){\omega }_D(x,d\xi )$, where ${\omega }_D$ is harmonic measure, the exit probability distribution of Brownian motion. The harmonic function equation is $\Delta u=0$, with the Laplace operator convention $\Delta ={\sum }_j{\partial }_j^2$.

Fix $x_0\in \partial \Omega$, $\epsilon >0$, and the power barrier for an exterior cone $g$ from part (i). Write $M=\Vert f{\Vert }_{\infty }$. By continuity of the boundary data, there is $\rho >0$ such that $|\phi (x)-\phi (x_0)|<\epsilon$ on $\partial \Omega \cap B_{\rho }(x_0)$. On the rest of the boundary, $-g\ge m{\rho }^{\nu }$. The boundary is compact, so $\phi$ is bounded. Choose $A>0$ large enough that $A\delta \ge M$ andThen the Perron subfunction for the Poisson equation and its superfunction counterpartsatisfy $\Delta v_{-}\ge M\ge f$, $\Delta v_{+}\le -M\le f$, and $v_{-}\le \phi \le v_{+}$ on the whole boundary. The near-boundary ordering uses the choice of $\rho$; the remaining ordering uses the choice of $A$. In particular the Perron family is nonempty and bounded above.

The Perron method for the Dirichlet problem and the permitted comparison give $v_{-}\le u\le v_{+}$. Since $g(x)\to 0$ as $x\to x_0$, we obtainLetting $\epsilon \downarrow 0$ provesThe interior Perron theorem gives continuity inside $\Omega$, and the displayed limit together with continuity of $\phi$ proves the continuous extension $u\in C^0(\overline{\Omega })$ with the required boundary trace. Thus every boundary point is a regular boundary point.

Fix $\epsilon >0$ and choose a boundary neighbourhood $U$ of $z$ on which $|\phi -\phi (z)|<\epsilon$. The positive continuous barrier $w$ has a positive minimum $m$ on the compact set $\partial \Omega \setminus U$. Choosing $A$ large enough makeson all of $\partial \Omega$. The left function is subharmonic and belongs to the Perron family; the right function is superharmonic and dominates every family member by part (ii). ThereforeAs $x\to z$, continuity gives $w(x)\to 0$. Letting $\epsilon \downarrow 0$ proves $\overline{u}(x)\to \phi (z)$. Such a $w$ is a barrier for the Dirichlet problem, and $z$ is a regular boundary point.

Potential theory studies harmonic functions, subharmonic functions, and the Laplace operator, including harmonic measure, regular boundary points, and the Dirichlet problem. The Brownian representation of the Dirichlet problem connects these analytic objects with exit probability distributions.