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Positive anchor values and the limits of uniform Harnack propagation

Statement

Let a family of positive Schrödinger solutions have a common finite chain cover joining a compact interior set $K$ to an anchor point $x_{*}$ , with common comparison constant $H$ . If $u (x_{*}) \geq a > 0$ uniformly, then $\inf_{K} u \geq a / H$ . A common limit $u \to c > 0$ on the end supplies such an anchor if the end threshold giving, for example, $u \geq c / 2$ is uniform in the family, with the comparison controls extending to the chosen anchor. A uniform end threshold is sufficient, but is not necessary for a uniform anchor. Individual end limits alone guarantee neither a fixed point with a positive uniform anchor value, nor a uniform end threshold, nor a uniform compact lower bound, even on a fixed Euclidean three-dimensional exterior with smooth nonnegative potentials uniformly bounded everywhere.

Facts & Assumptions

[F1]

The finite-chain comparison is the conditional part of Finite Harnack chains on fixed compact connected interior sets: common local pairwise comparisons on the half-balls of a common finite connected chain cover give a common comparison constant on the covered set. For a connected compact $K$ and an anchor outside $K$ , include a fixed path from $K$ to $x_{*}$ and cover its compact image together with $K$ ; a connected smooth manifold is path connected because chart balls are path connected and its path components are open. More generally, the assumed common connected chain cover must contain all of $K$ and $x_{*}$ . The coefficient and cover controls, when used to obtain the local comparisons, must hold on this enlarged compact region. Below, the examples' local comparisons are proved directly; the supplier's ZFC analytic consequence is not invoked.

[F2]

The exterior and Laplacian conventions are Weighted exterior elliptic domains and admissible weights. For the examples take $Σ = R^{3}$ with its Euclidean metric and empty compact boundary; outside the closed unit ball this is the prescribed exterior chart. Here $Δ u = \sum_{i = 1}^{3} \partial_{i}^{2} u$ . For a smooth positive $u$ , define $V = Δ u / u$ , so $(Δ - V) u = 0$ pointwise. No boundary condition is assumed in this lemma.

[F3]

Coordinatewise sum, product, quotient and chain differentiation use Sum, product, quotient, and chain rules from difference quotients. All functions differentiated below have strictly positive square-root arguments and denominators. Their potentials, signs, bounds and local comparisons are computed locally rather than imported from an external result.

Proof

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

Goal: Prove the anchor estimate and both distinctions concerning end normalization.

Proof technique: direct.

1.1F1givenalgebra

By [F1], $u (x_{*}) \leq H u (x)$ for every $x \in K$ . Insert $u (x_{*}) \geq a$ and take the infimum to obtain $\inf_{K} u \geq a / H$ . A common end threshold permits a single fixed $x_{*}$ beyond it, with $a = c / 2$ , provided the same comparison controls hold to this point. This proves the anchor estimate and the sufficiency assertion.

1.2F2F3algebra

For $0 < b < 1$ , $λ > 0$ and $z \in R^{3}$ , put $Q (x) = 1 + λ^{2} ∣ x - z ∣^{2}$ and $f (x) = 1 - b Q (x)^{- 1 / 2}$ . This is smooth and $f \geq 1 - b > 0$ . For $t > 0$ , rationalizing gives $(\sqrt{t + h} - \sqrt{t}) / h = 1 / (\sqrt{t + h} + \sqrt{t})$ for nonzero sufficiently small $h$ ; the quotient rule then gives $(t^{- 1 / 2})^{'} = - 1 / (2 t^{3 / 2})$ . Repeated differentiation on $t > 0$ gives smoothness. Applying [F3] coordinatewise yields $\partial_{i} f = b λ^{2} (x_{i} - z_{i}) Q^{- 3 / 2}$ and $\partial_{i}^{2} f = b λ^{2} Q^{- 3 / 2} - 3 b λ^{4} (x_{i} - z_{i})^{2} Q^{- 5 / 2}$ . Summing in [F2] yields $Δ f = 3 b λ^{2} Q^{- 5 / 2}$ , hence $0 \leq V_{f} : = Δ f / f \leq 3 b λ^{2} / (1 - b)$ . For fixed parameters, $f \to 1$ uniformly as $∣ x ∣ \to \infty$ , since $∣ x - z ∣ \geq ∣ x ∣ - ∣ z ∣$ .

2.1step 1.2F1algebra

To show that a uniform threshold is not necessary, let $e_{1} = (1, 0, 0)$ and, for integers $n \geq 1$ , set $w_{n} (x) = 1 - \frac{3}{4} (1 + ∣ x - 2 n e_{1} ∣^{2})^{- 1 / 2}$ . Step 1.2 gives positive smooth Schrödinger solutions with $0 \leq V_{w_{n}} \leq 9$ , and each has end limit $1$ . At the fixed anchor $0$ , $w_{n} (0) \geq 1 - 3 / (4 \sqrt{5}) > 1 / 2$ . Nevertheless $w_{n} (2 n e_{1}) = 1 / 4$ , so no radius works for the inequality $w_{n} \geq 1 / 2$ for every $n$ beyond that radius. The bounds $1 / 4 \leq w_{n} \leq 1$ give $w_{n} (x) \leq 4 w_{n} (y)$ for all $x, y$ , directly supplying common local comparisons on every fixed chain cover in [F1]. The Euclidean metric and the displayed potential bounds also give uniform coefficient controls there. Thus a common anchor can exist without the indicated uniform end threshold.

3.1step 1.2F1algebra∎

For $0 < ε \leq 1 / 2$ set $u_{ε} (x) = 1 - (1 - ε) (1 + ε^{2} ∣ x ∣^{2})^{- 1 / 2}$ . Step 1.2 gives $u_{ε} \geq ε > 0$ , $u_{ε} \to 1$ at infinity for each $ε$ , and $0 \leq V_{ε} \leq 3 ε (1 - ε) \leq 3 / 4$ everywhere. The Euclidean metric and these potential bounds give uniform coefficient controls on every fixed finite compact chain cover. To check the local comparisons directly, write $s = (1 + ε^{2} ∣ x ∣^{2})^{1 / 2} \geq 1$ . Then $u_{ε} (x) = ε + (1 - ε) ε^{2} ∣ x ∣^{2} / [s (s + 1)]$ . On $∣ x ∣ \leq R$ this implies $ε \leq u_{ε} (x) \leq ε (1 + R^{2} / 4)$ , using $s (s + 1) \geq 2$ and $ε \leq 1 / 2$ . Thus for all $x, y$ in this ball, $u_{ε} (x) \leq (1 + R^{2} / 4) u_{ε} (y)$ , uniformly in $ε$ , and every fixed finite chart cover with compact closures fits into such a ball. For the closed unit ball $K$ , $\inf_{K} u_{ε} = u_{ε} (0) = ε \to 0$ . Moreover the same bound gives $u_{ε} (x) \to 0$ at every fixed $x$ , so there is no fixed point with a positive uniform anchor value. For any proposed common end radius $R_{0}$ , fix $x$ with $∣ x ∣ > R_{0}$ ; then $u_{ε} (x) < 1 / 2$ for sufficiently small $ε$ . Hence the individual common end limits supply neither a uniform threshold nor a uniform compact lower bound, despite the common local comparison controls in [F1].

Depends on

Used by · 1

Exterior Schrödinger maximum principles, Hopf growth and Harnack

Dependency tree · 2 levels

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