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Smooth approximation of compactly supported interior heights

Statement

Work in ZFC. Let $(R, k)$ be a compact smooth Riemannian region (a manifold with boundary) of dimension $n \geq 0$ , and let $v : R \to R$ be Lipschitz with support compactly contained in its interior. There exist $v_{j} \in C_{c}^{\infty} (R^{\circ})$ , all supported in one fixed compact interior set, such that $∥ v_{j} - v ∥_{\infty} + \int_{R} (∣ v_{j} - v ∣ + ∣ d v_{j} - d v ∣_{k}) d V_{k} \to 0$ . Here $d v$ is the almost-everywhere weak differential, $∣ \cdot ∣_{k}$ is the dual metric norm and $d V_{k}$ is the Riemannian volume density; for $n = 0$ , differentials vanish and each point has volume one.

Lipschitz means that there is a single finite $L \geq 0$ such that $∣ v (p) - v (q) ∣ \leq L d_{k} (p, q)$ for every pair in the same connected component of $R$ , with the usual distance on $R$ . Here $d_{k}$ is the infimum of the lengths $L_{k} (γ) = \int ∣ \dot{γ} ∣_{k}$ of piecewise $C^{1}$ curves in $R$ joining the two points; distinct components have extended distance $+ \infty$ and impose no cross-component inequality.

Facts & Assumptions

[F1]

Finite smooth interior cutoffs form a partition equal to one near the compact support. Finite smooth cutoffs relative to compact sets and boundary neighborhoods

[F2]

Chart and metric first-Sobolev norms are uniformly comparable on each chosen compact chart; locally Lipschitz functions have coordinate-independent weak differentials, with the smooth-cutoff product rule. The supplier also fixes the dimension-zero conventions. Weak first derivatives and Riemannian chart norm comparison

[F3]

For every dimension $n \geq 1$ , a compactly supported Lipschitz coordinate function admits mollification converging uniformly and in $W^{1, 1}$ , with fixed compact support inside its coordinate domain; an admissible kernel exists. Compactly supported chart mollification in uniform and first Sobolev norms

[F4]

The metric is the smooth positive-definite tensor $k$ on $R$ . Riemannian metric and riemannian manifold

[F5]

Lipschitz continuity is the finite-constant metric inequality, applied on each connected component with one common bound. Intrinsic Riemannian distance is the infimum of piecewise $C^{1}$ curve lengths; the extended distance is $+ \infty$ between distinct components. Length is the integral of metric speed. Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction Riemannian distance on a connected manifold Extended riemannian distance on a disconnected manifold Riemannian speed and length

[F6]

On a compact set contained in a coordinate chart, the Riemannian quadratic form has uniform positive lower and finite upper bounds relative to the Euclidean quadratic form. In particular $k_{p} (ξ, ξ) \leq C ∣ ξ ∣^{2}$ there for some finite $C > 0$ . Local comparison of a riemannian metric with the euclidean metric

[F7]

Write $R^{\circ} = Int R$ for the manifold interior: in positive dimension its points have positive last boundary-chart coordinate; in dimension zero every point is interior and the boundary is empty. Interior and boundary of a manifold with boundary

[A1]

Work in ZFC, carrying the mollification supplier's ZFC hypothesis and the chart-comparison supplier's Countable Choice hypothesis. Choice is inherited through these analytic suppliers, not through the finite partition or the explicit sequence of radii. The Axiom of Choice

Proof

**Given:**The hypotheses in the Statement.

Goal: Establish the stated construction or identity.

1.1givenF1F2F3F5F6F7A1construct

If $v = 0$ , take $v_{j} = 0$ with empty support. If $n = 0$ , every point is interior by [F7], and every chart is a singleton, so $v$ is smooth; take $v_{j} = v$ , whose compact interior support is fixed. The zero-dimensional conventions in [F2] make both errors zero. Henceforth take $n \geq 1$ and nonempty support. Cover the support by finitely many interior coordinate balls $x_{a} : U_{a} \to B_{a}$ , each with closure contained in a larger interior chart, and choose the cutoffs of [F1]. The identity $v = \sum_{a} χ_{a} v$ holds globally, since $v$ vanishes outside its support. By [F6], on $\overline{U_{a}}$ the coordinate quadratic form satisfies $k_{p} (ξ, ξ) \leq C_{a}^{m e t} ∣ ξ ∣^{2}$ for a finite $C_{a}^{m e t} > 0$ . For $p, q \in U_{a}$ , the inverse-coordinate straight segment stays in $U_{a}$ and has length at most $\sqrt{C_{a}^{m e t}} ∣ x_{a} (p) - x_{a} (q) ∣$ by [F5]. Thus $d_{k} (p, q)$ is at most this bound, and the hypothesis gives $∣ v (p) - v (q) ∣ \leq L \sqrt{C_{a}^{m e t}} ∣ x_{a} (p) - x_{a} (q) ∣$ . In particular $v$ is locally Lipschitz in smooth coordinates, as required by [F2]. Its coordinate expression is also bounded on the bounded ball $B_{a}$ . Each coordinate cutoff is smooth with compact support in $B_{a}$ , so its derivative is bounded there and the coordinate straight-segment estimate makes it Lipschitz. The product inequality therefore shows that $f_{a} = (χ_{a} v) \circ x_{a}^{- 1}$ is Lipschitz and compactly supported in $B_{a}$ . Apply [F3] in dimension $n$ , using [A1], and fix for each chart a radius $r_{a} > 0$ such that every mollified support for radius at most $r_{a}$ stays in one fixed compact subset $K_{a}$ of that chart. Use the explicit radii $r_{a} / (j + 1)$ .

2.1F2F3A1step 1.1algebra∎

Pull the finitely many smooth coordinate approximants back to their charts, extend by zero, and sum them to obtain $v_{j}$ . The zero extensions are smooth because their supports stay away from the chart sides, and all sums are supported in the fixed compact interior union $⋃_{a} K_{a}$ . The supremum error is at most the sum of the finitely many coordinate supremum errors. Apply [F2] to each coordinate difference $f_{a, j} - f_{a}$ , with the same fixed compact support set $K_{a}$ for all $j$ ; linearity and the cutoff product rule identify the resulting summed weak differential with $d v_{j} - d v$ . The integral error is therefore at most $\sum_{a} C_{a} ∥ f_{a, j} - f_{a} ∥_{W^{1, 1}}$ , with constants $C_{a}$ independent of $j$ , and tends to zero by [F3]. Both stated convergences follow.

Depends on

Used by · 1

Boundary-constant-preserving smooth approximation

Dependency tree · 2 levels

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