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Harnack inequality on compact subsets

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ .

For $n \geq 2$ , a domain $Ω \subseteq R^{n}$ and a nonempty compact $K \subset Ω$ , there is $C = C (K, Ω, n) \geq 1$ such that every nonnegative harmonic function $u$ on $Ω$ satisfies $\sup_{K} u \leq C \inf_{K} u .$ The compact set $K$ need not be connected.

Facts & Assumptions

Given: Countable Choice and the objects and hypotheses in the statement (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Under Countable Choice, on $B_{R} (a)$ a nonnegative harmonic function satisfies $u (x) \leq (R / (R - ∣ x - a ∣))^{n} u (a)$ . (Harnack inequality on a ball).

[F2]

Any compact subset of a domain has a finite cover by interior balls with connected overlap graph and fourfold closed balls in the domain. (Finite harnack chain on a compact connected subset).

[F3]

A nonnegative harmonic function on a domain is identically zero or strictly positive everywhere. (Nonnegative harmonic function with an interior zero vanishes).

Proof

technique · direct

1.1F2F3given

If $u$ has an interior zero, it vanishes throughout $Ω$ , and the inequality is immediate. Otherwise it is strictly positive. Fix the finite admissible-ball family of the chain lemma, of size $N$ , independently of $u$ .

2.1F1step 1.1

If $x, y \in B_{r} (a)$ from this family, then $B_{3 r} (y) \subset B_{4 r} (a) \subset Ω$ and $∣ x - y ∣ < 2 r$ . Ball Harnack gives $u (x) \leq 3^{n} u (y)$ . Thus any two points of one of the small balls are comparable by the same constant.

3.1step 2.1algebra∎

For $x, y \in K$ , connect balls containing them by a simple path in the finite connected graph, with at most $N$ vertices. Choose one point in each of its finitely many overlaps and apply the local comparison in successive balls. It gives $u (x) \leq 3^{n N} u (y)$ . Taking the supremum in $x$ and the infimum in $y$ proves the claim with $C = 3^{n N}$ .

Depends on

Used by

Harnack convergence principle Theorem

Dependency tree · two levels

14 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

John K. Hunter, Notes on Partial Differential Equations (2014) (standard reference, not scraped)
Tsogtgerel Gantumur, Harmonic functions (2012) (standard reference, not scraped)