Perron method

The Perron method constructs a harmonic solution with prescribed boundary data as the pointwise supremum of all subharmonic functions lying below those data. Harmonic lifting proves interior harmonicity; barriers determine whether the supremum attains the boundary values.

A nonempty collection of real continuous subharmonic functions on a connected Riemann surface is a Perron family if it is closed under pairwise maxima and harmonic lifting of a continuous subharmonic function on every relatively compact coordinate disk. Its pointwise supremum is either everywhere infinite or a harmonic function on a Riemann surface. At a point where the supremum is finite, take increasing disk lifts approaching that value. The Harnack inequality for harmonic functions gives a finite harmonic limit on the disk. Lifting their maxima with any other member gives a larger harmonic limit with the same value at the chosen point, hence the same limit by the strong maximum principle for harmonic functions. The supremum is therefore harmonic on that disk; connectedness propagates finiteness to the whole surface.

A continuous function $v$ is a Perron subfunction for $\Delta u=f$ when it lies below every classical solution of that Poisson equation on a relatively compact open ball whenever it lies below that solution on the ball's boundary. For $C^2$ functions, this is equivalent to $\Delta v\ge f$. A superfunction reverses the comparison and has $\Delta v\le f$ when $C^2$. Adding a boundary upper bound gives the family whose supremum defines the Perron method for the Dirichlet problem.

A barrier at $z\in \partial \Omega$ is a continuous superharmonic function that vanishes at $z$ and is strictly positive at every other point of the closure. It forces the Perron solution to converge to its prescribed boundary value at $z$.

A boundary point is regular for the Dirichlet problem when the Perron solution approaches the prescribed continuous boundary value there. The existence of a barrier implies regularity.

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 planar set outside a global solid cone of half-angle $\alpha$, choose $0<{\alpha }_0<\min (\alpha ,\pi /2)$, $\beta =\pi /[2(\pi -{\alpha }_0)]$, and $0<\nu <\beta$. If $\psi$ is the angle from the opposite axis, the Laplacian in polar coordinates givesThe cosine has a positive uniform lower bound outside the original cone. On a bounded set, $\Delta g$ therefore has a positive uniform lower bound as well. The positive function $-g$ is a barrier for the Dirichlet problem. An unbounded set does not receive the same global lower bound from this construction.

A domain satisfies the exterior sphere condition at a boundary point when a ball contained in the complement is tangent to the boundary only at that point. Fundamental solutions of the Laplace equation centred at the ball's centre provide a harmonic barrier, so such a point is regular.

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.