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Finite Harnack chains on fixed compact connected interior sets

Statement

Let $(Σ, h)$ be a smooth three-dimensional Riemannian exterior and let $K \subset Σ^{\circ}$ be a nonempty compact connected subset, with compactness and connectedness taken in its subspace topology. There are finitely many interior coordinate balls $U_{i}$ , $1 \leq i \leq N$ , with compact closures in their chart domains, whose concentric half-balls $H_{i}$ cover $K$ . Each $H_{i}$ meets $K$ , and the graph with vertices $1, \dots, N$ and an edge between distinct $i, j$ exactly when $H_{i} \cap H_{j} \cap K \neq \emptyset$ is connected, meaning any two vertices can be joined by a finite sequence of edges. All balls are Euclidean in their respective coordinates. For any positive real function $u$ on $⋃_{i} U_{i}$ , if $C_{*} \geq 1$ and $u (x) \leq C_{*} u (y)$ for every $x, y \in H_{i}$ and every $i$ , then $\sup_{K} u \leq C_{*}^{N} \inf_{K} u$ . Thus the conditional comparison applies to every positive solution of $(Δ_{h} - V) u = 0$ for which these local comparisons hold.

In particular, work in ZFC for the following analytic consequence. Fix such a finite chart cover. Consider smooth Riemannian metrics $h$ , potentials $V : ⋃_{i} U_{i} \to [0, \infty)$ bounded on each $U_{i}$ , and positive real $C^{2}$ functions $u$ solving $(Δ_{h} - V) u = 0$ pointwise there, where $Δ_{h} = {div}_{h} {grad}_{h}$ . Write $r_{i}$ for the coordinate radius of $U_{i}$ and, in its coordinates, put $m_{i} = \sqrt{\det (h_{a b})}$ , $A_{i} = (m_{i} h^{a b})$ and $q_{i} = m_{i} V$ . Suppose fixed constants $λ_{i} > 0$ , $M_{i} < \infty$ and $Q_{i} \geq 0$ , independent of the family member, satisfy $ξ^{T} A_{i} (x) ξ \geq λ_{i} ∣ ξ ∣^{2}$ , $∥ A_{i} (x) ∥_{o p} \leq M_{i}$ and $q_{i} (x) \leq Q_{i}$ on $U_{i}$ for every $ξ \in R^{3}$ . Then the comparison is uniform over the family: $C_{*}$ can be chosen using only the finitely many parameters $3$ , $M_{i} / λ_{i}$ and $r_{i}^{2} Q_{i} / λ_{i}$ . No measurability or smoothness of $V$ beyond these hypotheses is required. The preceding conditional chain comparison does not impose a sign condition on $V$ when the local comparisons are supplied independently.

Facts & Assumptions

[F1]

Compact subsets are compact in their subspace topology by Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right: every relatively open cover has a finite subcover. Connected subsets admit no partition into two disjoint nonempty relatively open sets by Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets.

[F2]

The smooth exterior, its interior charts, the Riemannian metric and the Laplacian convention are those of Weighted exterior elliptic domains and admissible weights. The letter $K$ here denotes the subset in the Statement, not necessarily the compact core of that definition. The sets $U_{i}, H_{i}$ and their graph are defined in the Statement; they are not required to be metric balls.

[F3]

Multiplicative interior Harnack for geometric Schrödinger solutions applies to a nonnegative real $C^{2}$ pointwise solution on a coordinate ball for a smooth metric and a bounded nonnegative potential, with the displayed bounds on $A_{i}, q_{i}$ . It gives a half-ball comparison constant at least one depending only on dimension, $M_{i} / λ_{i}$ and $r_{i}^{2} Q_{i} / λ_{i}$ . The analytic consequence inherits its ZFC assumption, including The Axiom of Choice; that imported assumption is used precisely when this supplier is invoked in step 3.1. The finite-cover and finite-chain arguments themselves require no arbitrary-index choice.

Proof

Given: The data and hypotheses in the Statement and [F1]–[F3].

Goal: The finite connected cover, conditional chain comparison and stated uniform analytic consequence.

Proof technique: direct.

1.1givenF1F2

Consider all interior chart balls centered at points of $K$ with compact closures inside their chart domains, together with their concentric half-balls. Such balls exist around every point by shrinking an interior coordinate neighborhood. The traces of these half-balls on $K$ are a relatively open cover of $K$ , so [F1] supplies a finite subcover; retain the associated balls and discard any empty traces. Write $U_{i}, H_{i}$ , $1 \leq i \leq N$ , for the resulting balls and half-balls. If their graph were disconnected, fix a vertex and let $I$ consist of the vertices reachable from it by finite edge sequences. Then $I$ and its complement are both nonempty, and there is no edge between them. The unions of $H_{i} \cap K$ over these two index sets are nonempty, relatively open, disjoint, and cover $K$ . This contradicts connectedness in [F1], proving the geometric assertion.

2.1givenstep 1.1algebra

For $x, y \in K$ , choose half-balls containing them. By step 1.1 their vertices are joined by a finite edge sequence; deleting segments between repeated vertices gives a path of $s \leq N$ distinct vertices. At each edge take a point in the corresponding intersection with $K$ . The resulting sequence starts at $x$ , ends at $y$ , and each consecutive pair lies in one of the $s$ half-balls. Applying the assumed local comparisons and multiplying gives $u (x) \leq C_{*}^{s} u (y) \leq C_{*}^{N} u (y)$ , also when $s = 1$ . Taking the supremum in $x$ and then the infimum in $y$ proves the conditional bound. These extrema are finite and positive as bounded by any fixed value through the pairwise inequalities; continuity is not needed for this conditional argument.

3.1givenF2F3step 2.1∎

For the analytic family, on each fixed $U_{i}$ the metric is smooth, $V$ is bounded and nonnegative, $u$ is positive and $C^{2}$ , and the pointwise equation and all three coefficient bounds are exactly those of [F3], with dimension three. Thus [F3] supplies a constant $C_{i} \geq 1$ using only $3$ , $M_{i} / λ_{i}$ and $r_{i}^{2} Q_{i} / λ_{i}$ , common to all family members. Take $C_{*} = \max_{1 \leq i \leq N} C_{i}$ and apply step 2.1. Both $N$ and these coefficient parameters are fixed, so the resulting comparison is uniform as asserted.

Depends on

Used by · 2

Dependency tree · 2 levels

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