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A harmonic function is recovered from its values on any containing circle by the Poisson formula

Statement

Let $u$ be harmonic on an open set containing the closed disc $\overline{D (a, R)}$ , and let $z = a + ρ e^{i ϕ}$ with $0 \leq ρ < R$ . Then

$u (z) = \frac{1}{2 π} \int_{0}^{2 π} \frac{R^{2} - ρ^{2}}{R^{2} - 2 R ρ \cos (ϕ - t) + ρ^{2}} u (a + R e^{i t}) d t .$

So a harmonic function on a disc is recovered from its boundary values on any larger concentric circle lying inside its domain.

Facts & Assumptions

Given: A harmonic function $u$ on a neighbourhood of $\overline{D (a, R)}$ .

[L1]

The Poisson integral gives the unique continuous harmonic extension from the unit-circle boundary to the closed unit disc (The Poisson integral gives the unique continuous harmonic extension on the closed unit disc).

[L2]

Two continuous harmonic functions on the same bounded domain with the same boundary values agree (The bounded plane Dirichlet problem has at most one continuous harmonic solution).

Proof

technique · direct

1.1givenalgebra

Define $v (w) : = u (a + R w)$ on $\overline{D}$ . Then $v$ is continuous on $\overline{D}$ , harmonic on $D$ , and has boundary values $v (e^{i t}) = u (a + R e^{i t})$ .

2.1step 1.1L1L2

By [L1], the Poisson integral of the boundary function $t \mapsto u (a + R e^{i t})$ is a continuous harmonic function on $\overline{D}$ with the same boundary values as $v$ ; [L2] therefore makes it equal to $v$ throughout $\overline{D}$ .

3.1step 2.1algebra∎

Evaluating step 2.1 at $w = (z - a) / R = (ρ / R) e^{i ϕ}$ gives exactly the displayed formula, because the unit-disc kernel there is $\frac{1 - (ρ / R)^{2}}{1 - 2 (ρ / R) \cos (ϕ - t) + (ρ / R)^{2}} = \frac{R^{2} - ρ^{2}}{R^{2} - 2 R ρ \cos (ϕ - t) + ρ^{2}} .$

Depends on

Used by

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sigurdur Helgason, MIT 18.112 Lecture 16: Harmonic Functions (standard reference, not scraped)