the Perron method
/ peh-ROHN /
The Poisson formula solves the Dirichlet problem on a disk, and conformal mapping carries that solution to any region you can map onto a disk. But many regions are too rough or topologically awkward to map by a clean formula. The Perron method is the elegant, formula-free way to solve the Dirichlet problem on a remarkably general region: it builds the harmonic solution as the least upper envelope of all the subharmonic functions that stay below the boundary data.
Here is the method walked through. Suppose D is a bounded region and g is a continuous function on its boundary. Form the Perron family: all subharmonic functions v on D whose boundary limsup is at most g at every boundary point. This family is rich (it contains at least the constant equal to the minimum of g) and flexible (the maximum of two members is again a member, and you can replace a member by its harmonic 'lift' over any disk). Now define u(z) = the supremum of v(z) over all v in the family. The miracle, proved using Harnack's principle, is that this pointwise supremum u is automatically harmonic inside D — the subharmonic competitors push up against an invisible harmonic ceiling, and u is that ceiling. So existence of a harmonic interior function is settled in one stroke, for essentially any region.
The Perron method cleanly separates two jobs that used to be tangled. Getting a harmonic function inside is the easy part and works for any bounded region. The hard, genuinely geometric part is whether u actually attains the boundary value g continuously at a given boundary point; that succeeds precisely at the regular boundary points, the ones that possess a barrier. The honest summary: Perron always produces a harmonic interior candidate, but whether it solves the Dirichlet problem at the boundary (takes the right values there) can fail at irregular points like an isolated puncture or an inward cusp — and detecting those points is what barriers are for.
On the punctured disk 0 < |z| < 1 with boundary data 0 on the outer circle and 1 at the puncture z = 0, the Perron method yields the harmonic interior solution u identically 0 — it ignores the single isolated puncture, which is an irregular (removable) boundary point with no barrier.
Perron builds a harmonic interior even here, but the value at the isolated puncture is not attained — that point is irregular.
Perron is named for Oskar Perron. Its strength is decoupling existence (always) from boundary attainment (only at regular points); it does not by itself tell you which boundary points are regular — for that you must exhibit a barrier.