Math 213a: Complex analysis Problem Set #7 (5 November 2003): Harmonic functions and the Dirichlet problem

1. In Ahlfors’ first exercise for V.1.3 (p.183) he gives a geometric description of the harmonic function P_{U}(z) on the open unit disc Δ obtained from Poisson’s integral formula when U is the characteristic function of an arc on the unit circle. Explain this result using a suitable conformal map from Δ to an infinite strip.

Some more problems from Ahlfors (V.1.4, p.184; V.2.1, p.196)

2. If E is a compact subset of a region Ω, prove that there exists a constant M, depending only on E and Ω, such that every positive harmonic function u : Ω → (0, ∞) satisfies the inequality u(z′) ≤ Mu(z) for all z, z′ ∈ E.

3. Show that the functions |x| = | Re(z)|, |z|^{α} (all α ≥ 0), and log(1 + |z|^{2}) are subharmonic.

4. If v is a C^{2} function on some region, prove that v is subharmonic if and only if Δv ≥ 0. [See Ahlfors for a hint.]

5. Extend Harnack’s principle [Ahlfors V.1.4, Thm.6, 183–4] to subharmonic functions. (Warning: the conclusion must be weaker, as evidenced by such counterexamples as Ω_{n} = Ω = C, u_{n} = n|z|.)

The final two problems concern a discrete analogue of the Laplacian. We work on the cubic lattice L = Z^{n} ∈ R^{n}, regarded as an infinite graph of degree 2n (so z, z′ ∈ L are adjacent iff |z′ − z| = 1). An “interior point” of a subset S ∈ L is a point all of whose neighbors are in S; these points constitute the “interior” of S, whose complement in S is the “boundary” ∂S of S. A function u : S → R is harmonic if its value at each interior point z ∈ S is the average of its values at the neighbors of z, and subharmonic if u(z) ≤ (2n)^{−1} ∑_{|z′−z|=1} u(z′) for all interior z ∈ S. We similarly define (sub)harmonic functions on cL for any c > 0.

6. Suppose S is finite. Prove that any function U : ∂S → R extends to a unique harmonic function u : S → R, which is nonnegative if U is.

7.^{∗} Suppose K ∈ R^{n} is a convex compact set and v is a harmonic function on some neighborhood of K. For c > 0 let S_{c} = cL ∩ K, and let u_{c} be the harmonic function on S_{c} such that u_{c}(z) = v(z) for all z ∈ ∂S_{c} (this is welldefined, by the previous problem). Prove that there exists a constant A, depending only on K and u, such that |u_{c}(z) − v(z)| ≤ Ac^{2} for all z ∈ S_{c}. You may assume the theorem that v is automatically real-analytic; we didn’t prove this in class for n ≥ 3, but it follows easily from the Poisson integral representation of a harmonic function on a closed sphere in R^{n}.

[This is the beginning of one approach to justifying the numerical approximation of solutions of the Dirichlet problem on R^{n} by discrete harmonic functions.]

This problem set is due Wednesday, November 12, at the beginning of class.