a barrier and a regular boundary point
The Perron method always cooks up a harmonic function inside a region, but whether that function actually arrives at the prescribed boundary value as you approach a given edge point can fail. A barrier is a small certificate, built at a single boundary point, that guarantees success there. A boundary point that has a barrier is called a regular boundary point; the solution then takes the right boundary value continuously at that point.
Concretely, a barrier at a boundary point zeta of a region D is a function w defined near zeta inside D that is superharmonic (its negative is subharmonic), strictly positive everywhere near zeta except at zeta itself where it tends to 0 as you approach. Think of it as a little tent pole anchored to the ground exactly at zeta: it lets you squeeze the Perron solution from above and below so that, as z -> zeta, u(z) is forced toward the boundary value g(zeta). If such a w exists, zeta is regular and the Dirichlet problem is solved there; if no barrier exists, zeta is irregular and the solution may refuse to attain g(zeta). Whether a barrier exists is a purely local, geometric question about how the boundary looks near zeta.
Two facts make this practical. First, regularity is decided by the shape of the boundary: any boundary point you can touch with the tip of a line segment (or any non-degenerate continuum) reaching out of D is regular — so all reasonable boundaries (smooth curves, polygons, anything with a bit of bulk at each point) are entirely regular, and the Dirichlet problem is fully solvable. Second, the failures are genuinely exotic: an isolated puncture in the plane is the classic irregular point (a single point is too thin to support a barrier), and inward-pointing cusps can also misbehave. The takeaway: for the regions you meet in physics and geometry the boundary is regular everywhere and Perron solves the problem completely; barriers are the precise tool that draws the line between the well-behaved boundaries and the pathological ones.
At a smooth boundary point of a disk, the function w(z) = (1 - Re(z / zeta)) near the boundary point zeta on the unit circle is positive inside, vanishes at zeta, and is harmonic hence superharmonic — a barrier, certifying every disk boundary point as regular.
A simple barrier at a circle's boundary point, confirming the disk's boundary is everywhere regular.
Regularity is a property of the boundary point and the region, not of the boundary data g; an irregular point (like an isolated puncture) stays irregular no matter what values you prescribe — the failure is geometric, not about the data.