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 PU(z) on the open unit disc ∆ obtained from

Poisson’s integral formula when Uis the characteristic function of an arc

on the unit circle. Explain this result using a suitable conformal map

from ∆ to an infinite strip.

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

2. If Eis a compact subset of a region Ω, prove that there exists a constant M,

depending only on Eand Ω, such that every positive harmonic function

u: Ω →(0,∞) satisfies the inequality u(z0)≤Mu(z) for all z , z0∈E.

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

subharmonic.

4. If vis a C2function on some region, prove that vis subharmonic if and only

if ∆v≥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 Ωn=Ω=C,un=n|z|.)

The final two problems concern a discrete analogue of the Laplacian. We work

on the cubic lattice L=Zn∈Rn, regarded as an infinite graph of degree 2n

(so z, z0∈Lare adjacent iff |z0−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 Sis the “boundary” ∂S of S. A function u:S→R

is harmonic if its value at each interior point z∈Sis the average of its values

at the neighbors of z, and subharmonic if u(z)≤(2n)−1P|z0−z|=1 u(z0) for all

interior z∈S. We similarly define (sub)harmonic functions on cL for any c > 0.

6. Suppose Sis finite. Prove that any function U:∂S →Rextends to a unique

harmonic function u:S→R, which is nonnegative if Uis.

7.∗Suppose K∈Rnis a convex compact set and vis a harmonic function on

some neighborhood of K. For c > 0 let Sc=cL ∩K, and let ucbe the

harmonic function on Scsuch that uc(z) = v(z) for all z∈∂Sc(this is well-

defined, by the previous problem). Prove that there exists a constant A,

depending only on Kand u, such that |uc(z)−v(z)| ≤ Ac2for all z∈Sc.

You may assume the theorem that vis 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 Rn.

[This is the beginning of one approach to justifying the numerical approximation

of solutions of the Dirichlet problem on Rnby discrete harmonic functions.]

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

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

In Ahlfors’ first exercise for V.1.3 (p.183) he gives a geometric description of the harmonic function PU (z) on the open unit disc ∆ 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 ∆ to an infinite strip.

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

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′) ≤ M u(z) for all z, z′^ ∈ E.
Show that the functions |x| = | Re(z)|, |z|α^ (all α ≥ 0), and log(1 + |z|^2 ) are subharmonic.
If v is a C^2 function on some region, prove that v is subharmonic if and only if ∆v ≥ 0. [See Ahlfors for a hint.]
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 Ωn = Ω = C, un = n|z|.)

The final two problems concern a discrete analogue of the Laplacian. We work on the cubic lattice L = Zn^ ∈ Rn, 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.

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 ∈ Rn^ is a convex compact set and v is a harmonic function on some neighborhood of K. For c > 0 let Sc = cL ∩ K, and let uc be the harmonic function on Sc such that uc(z) = v(z) for all z ∈ ∂Sc (this is well- defined, by the previous problem). Prove that there exists a constant A, depending only on K and u, such that |uc(z) − v(z)| ≤ Ac^2 for all z ∈ Sc. 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 Rn.

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

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

Complex Algebra 8 - Exercises - Mathematics, Exercises of Mathematics

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