Perron’s Method and Barriers

Book 6A solved the Dirichlet problem for the Laplacian in two ways: by Poisson's formula on a ball (6A.2 Harmonic Functions, 6B.4 Green’s Functions, Separation of Variables and Transforms) and by Hilbert-space methods on any bounded domain, in the weak sense (6A.6 Weak Solutions and Elliptic Regularity). Neither answers the most classical form of the question: given a bounded domain $\Omega$ and a continuous function $\varphi$ on its boundary, is there a function harmonic in $\Omega$, continuous on $\overline\Omega$, equal to $\varphi$ on $\partial\Omega$? The weak solution lies in $H^1$ and has boundary values only in the trace sense; and a continuous $\varphi$ need not be the trace of any $H^1$ function at all.

Oskar Perron's method (1923) answers the question with nothing but the maximum principle. It builds the solution as the supremum of all subharmonic functions lying below the data, shows that this supremum is harmonic, and then asks separately whether it takes the right boundary values. The answer to the second question depends on the geometry of the boundary, through barriers, and it can fail: at an isolated boundary point, or at the tip of a sufficiently sharp inward spike. The method separates the two halves of the problem as cleanly as possible: interior existence is automatic; boundary behaviour is local and geometric.

The same chapter treats the discrete Dirichlet problem, where the maximum principle again gives existence, and proves that finite-difference solutions converge at the rate $h^2$. Barriers and comparison functions are the everyday tools of elliptic and parabolic theory, and the barrier built from the distance to the boundary (Exercise 6.19) is the same computation as the Laplacian comparison theorem of Riemannian geometry (9B.1 Laplacian Comparison).

By the end of this chapter you will be able to:

• define subharmonic functions without derivatives, and prove their maximum and comparison principles and the harmonic lifting construction;
• prove Harnack's convergence theorem, and use it to prove that Perron's solution is harmonic;
• define barriers and regular boundary points, prove that a barrier gives the right boundary values, and construct barriers from exterior spheres;
• explain why the punctured disc and Lebesgue's spine have irregular boundary points, and state Wiener's criterion;
• set up the discrete Dirichlet problem, prove its maximum principle, and prove $O(h^2)$ convergence of the five-point scheme.

6.1Random walks and electrical networks

6.2Subharmonic functions without derivatives

In 6A.2 Harmonic Functions a $C^2$ function with $\Delta u \geq 0$ was called subharmonic, and shown to lie below its averages. Perron's method needs a notion that survives taking suprema and maxima, so it is defined by comparison with harmonic functions instead.

In words: a subharmonic function lies below the harmonic function with the same boundary values on any ball. On a ball the harmonic function with given continuous boundary values exists and is given by Poisson's integral (6B.4 Green’s Functions, Separation of Variables and Transforms), so the definition can be checked.

The definition is local in the following sense: a continuous function satisfying the sub-mean-value inequality (2) on all small spheres around each point is subharmonic (Exercise 6.14). So $|x|$, $\max(x_1, 0)$ and the maximum of two harmonic functions are subharmonic, although none is $C^2$.

Harmonic lifting. The key construction raises a subharmonic function on a ball to the harmonic function with the same boundary values there.

Harnack's convergence theorem. The other ingredient is a compactness statement for monotone sequences of harmonic functions, which follows from the Harnack inequality of 6A.2 Harmonic Functions.

6.3Perron's method

Let $\Omega$ be bounded and $\varphi$ continuous on $\partial\Omega$. A continuous function $v$ on $\overline\Omega$ is a subfunction for $\varphi$ if it is subharmonic in $\Omega$ and $v \leq \varphi$ on $\partial\Omega$. Let $S_\varphi$ be the set of subfunctions. It is not empty (the constant $\min\varphi$ belongs to it), and every $v \in S_\varphi$ satisfies $v \leq \max\varphi$, by the maximum principle. So

is a well-defined bounded function on $\Omega$, the Perron solution. If the Dirichlet problem has a solution $w$, then $w \in S_\varphi$, and $v \leq w$ for every $v \in S_\varphi$ by comparison, so $u = w$: the Perron solution is the only possible candidate.

The proof never asked how $u$ behaves at the boundary, and it can't: that depends on the boundary.

6.4Barriers and regular points

The proof uses only the barrier's sign and its superharmonicity: the barrier is a scaffold that pins $u$ near $\varphi(\xi)$ from above and below. It is a local property: a barrier defined only on $\overline\Omega\cap B(\xi, r)$ can be extended to a global one (Gilbarg–Trudinger, section 2.8), so regularity is decided by the shape of the boundary near $\xi$.

Exterior spheres. The simplest source of barriers is a ball outside $\Omega$ touching $\xi$: suppose there is a closed ball $\overline{B(y, R)}$ with $\overline{B(y, R)}\cap\overline\Omega = \{\xi\}$ (the exterior sphere condition). Then

is harmonic away from $y$, vanishes at $\xi$ (where $|x - y| = R$) and is positive at all other points of $\overline\Omega$ (where $|x - y| > R$): a barrier. Every bounded domain with $C^2$ boundary, and every convex domain, satisfies the exterior sphere condition at each boundary point (for convex domains use a ball beyond a supporting plane, Exercise 6.15). So the Dirichlet problem is solvable for continuous data on every bounded convex domain and every bounded domain with $C^2$ boundary. A weaker condition suffices: an exterior cone with vertex at $\xi$ (Poincaré's and Zaremba's condition) gives a barrier too, so Lipschitz domains, including polygons and polyhedra, are fine (Gilbarg–Trudinger, problem 2.12; Jost, section 4.4).

6.5Where the Dirichlet problem fails

In the plane, isolated points are the only kind of problem for domains whose complement has no tiny pieces: if each boundary point lies in a connected piece of the complement containing more than one point, it is regular. In three dimensions, a boundary can fail to be regular even at a point of a solid, connected complement.

The exact criterion was found by Norbert Wiener in 1924. For $n \geq 3$ and $\lambda \in (0, 1)$, let $E_k = \{x \notin \Omega : \lambda^{k+1} \leq |x - \xi| \leq \lambda^k\}$ be the pieces of the complement in dyadic-type shells around $\xi$. Then:

Capacity measures how much a set matters to potential theory: a ball of radius $\rho$ has capacity proportional to $\rho^{n-2}$, a point has capacity zero, and a thin spike of radius $\rho(t)$ has capacity per unit length about $\frac{1}{\log(1/\rho)}$. The criterion says the complement must be "thick enough at all scales" near $\xi$: the sum compares the capacity of each piece with that of the shell. For the exterior cone each term is comparable to a constant, so the sum diverges; for Lebesgue's spike with radius $e^{-1/|x_3|}$ the terms decay fast enough that it converges. Taylor II, §11.6 (Newtonian capacity), develops capacity and its probabilistic meaning.

6.6The discrete Dirichlet problem

Perron's method is an existence proof, not an algorithm. The algorithm is to discretise. On a grid of spacing $h$, replace $\Delta$ by the five-point Laplacian (in two dimensions; $2n + 1$ points in $n$)

Let $\Omega_h$ be the grid points in $\Omega$ whose $2n$ neighbours are all in $\overline\Omega$ (interior points; for simplicity, assume the boundary passes through grid points, as for a rectangle), and $\partial\Omega_h$ the grid points on $\partial\Omega$ that are neighbours of interior points. The discrete Dirichlet problem is: $\Delta_hu = f$ on $\Omega_h$, $u = \varphi$ on $\partial\Omega_h$. Note that $\Delta_hu(x) = 0$ says exactly that $u(x)$ is the average of its $2n$ neighbours: discrete harmonic, as in the electrical network.

Existence by uniqueness: the finite-dimensional Fredholm alternative (6A.6 Weak Solutions and Elliptic Regularity). The matrix is symmetric, negative definite and sparse, and it is solved in practice by iterative methods; the simplest, Jacobi iteration, replaces each interior value by the average of its neighbours, over and over, which is exactly the relaxation that computed 6A.2 Harmonic Functions's saddle figure. It converges, slowly, because the iteration matrix has spectral radius less than one (its largest eigenvalue is about $1 - \frac{\pi^2h^2}{2}$ on a unit square), and multigrid methods accelerate it enormously.

The proof has the structure of every convergence proof for numerical methods: consistency (the exact solution almost satisfies the scheme) plus stability (the scheme does not amplify errors) gives convergence. Here stability is the discrete maximum principle; in 6D.9 Finite Difference Methods it is the same principle for the heat equation, and the Lax equivalence theorem makes "consistency + stability = convergence" general. Jost, section 4.1, proves that solutions of the discrete problem converge to the solution of the continuous Dirichlet problem under much weaker assumptions, which gives another existence proof.

Schwarz's alternating method (1870) is a third route to existence, and an ancestor of modern domain decomposition: to solve the Dirichlet problem on the union of two overlapping domains where it is solvable separately (say a disc and a rectangle), solve on the first with the current guess as boundary data on the part of its boundary inside the second, then on the second, and alternate; the maximum principle shows the iterates converge (Jost, section 4.3).

6.7History

Hermann Amandus Schwarz introduced his alternating method in 1870. Henri Poincaré used sweeping out (balayage), a precursor of Perron's method, in 1887–90, and proved solvability under an exterior cone condition. Stanisław Zaremba gave the punctured disc example in 1911, and Henri Lebesgue his spine in 1913. Oskar Perron's method was published in 1923, and Norbert Wiener's criterion in 1924. Shizuo Kakutani's representation of harmonic functions by Brownian motion appeared in 1944. Richard Courant, Kurt Friedrichs and Hans Lewy's 1928 paper on difference equations proved convergence of finite-difference schemes and introduced the CFL condition (6D.9 Finite Difference Methods). Doyle and Snell's book appeared in 1984.

6.8Exercises

Solution

$u - h$ is continuous on $\overline B$ and satisfies the local sub-mean-value property in $B$ (since $h$ has the exact mean value property). If $\max_{\overline B}(u - h) = M > 0$, it is attained in $B$ (it is $\leq 0$ on $\partial B$); the set where $u - h = M$ is closed in $B$ and, by the local sub-mean-value property, open, so it is all of $B$, and by continuity $u - h = M > 0$ on $\partial B$, a contradiction.

Solution

(a) Every point $x$ of the closed ball satisfies $a\cdot(x - \xi) \geq 0$, with equality only at $x = \xi$ (the ball lies on the far side of the hyperplane and touches it only at $\xi$). Points of $\overline\Omega$ satisfy $a\cdot(x - \xi) \leq 0$. So the only common point is $\xi$. (b) Near each boundary point the boundary is the graph of a $C^2$ function with bounded second derivatives, say by $K$, uniformly over the compact boundary; a ball of radius less than $1/K$ tangent to the boundary from outside lies on the outer side of the graph and touches it only at the tangency point.

Solution

$u - v$ is harmonic and bounded on the annulus $\delta < |x| < \frac12$, zero on $|x| = \frac12$. The function $\varepsilon\log\frac{1}{2|x|}$ is harmonic, zero on $|x| = \frac12$ and at least $\sup|u - v|$ on $|x| = \delta$ once $\delta$ is small. By comparison $|u - v| \leq \varepsilon\log\frac{1}{2|x|}$; let $\delta \to 0$ and then $\varepsilon \to 0$. So $u = v$ away from $0$, and $v$ is harmonic on the whole disc of radius $\frac12$.

Solution

Interior points $(1, 1), (2, 1), (1, 2), (2, 2)$. By symmetry in $y \leftrightarrow 3 - y$, $u(1, 1) = u(1, 2) = a$ and $u(2, 1) = u(2, 2) = b$. Averaging: $4a = 1 + b + a + 0$ (neighbours $(0, 1)$, $(2, 1)$, $(1, 2)$, $(1, 0)$), so $3a = 1 + b$; and $4b = a + 0 + b + 0$, so $3b = a$. Hence $a = \frac38$, $b = \frac18$. All values lie between $0$ and $1$.

Solution

$\sin(\pi(x + h)) + \sin(\pi(x - h)) - 2\sin\pi x = -2(1 - \cos\pi h)\sin\pi x$, and similarly with $\sinh$ and $\cosh\pi h - 1$. Expanding, $\frac{2(1 - \cos\pi h)}{h^2} = \pi^2 - \frac{\pi^4h^2}{12} + O(h^4)$ and $\frac{2(\cosh\pi h - 1)}{h^2} = \pi^2 + \frac{\pi^4h^2}{12} + O(h^4)$, so $\tau = \Delta_hu = \frac{\pi^4h^2}{6}u + O(h^4)$ (since $\Delta u = 0$). The bound gives $\frac{2h^2}{12}\pi^4|u|$ because $\partial_x^4u = \pi^4u$ and $\partial_y^4u = \pi^4u$: it is attained to leading order. The error is genuinely of order $h^2$: $\partial_x^4u + \partial_y^4u = 2\pi^4u \neq 0$ although $\Delta u = 0$.

Solution

(a) $d = R - r$ with $r = |x|$: $\Delta d = -\Delta r = -\frac{n - 1}{r}$. The formula gives $-(n - 1)\frac{1/R}{1 - (R - r)/R} = -(n - 1)\frac{1/R}{r/R} = -\frac{n - 1}{r}$. (b) With $\kappa_i \geq 0$ and $\kappa_id < 1$ (which holds near the boundary), each term $\frac{\kappa_i}{1 - \kappa_id}$ is $\geq 0$, so $\Delta d \leq 0$: $d$ is superharmonic there. It is positive in $\Omega$ and vanishes on the whole boundary, so it is a barrier for the entire boundary at once near $\partial\Omega$; this is how boundary gradient estimates are proved for equations on convex domains (6C.6 Quasilinear Elliptic Equations and Minimal Surfaces). (c) The Laplacian of a distance function equals the mean curvature of its level set (the sum of the principal curvatures of the parallel hypersurface, with the sign convention that spheres seen from inside have positive curvature): for $|x|$ the level sets are spheres of radius $r$ with mean curvature $\frac{n - 1}{r}$; on a manifold with $\operatorname{Ric} \geq 0$, distance spheres are at most as curved as Euclidean ones, which is the comparison $\Delta r \leq \frac{n - 1}{r}$.