PropositionStatement: Literature-sourcedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Pointwise and energy difference estimates for odd powers

PDF p. 15: pointwise and gradient difference bounds in Proposition 3.9

Statement

Assume the Axiom of Choice. Let $n \geq 2$ , let $Θ \subset R^{n}$ be nonempty and compact with smooth boundary, and suppose that $Ω = R^{n} ∖ Θ$ is connected. Let $m \in {3, 5}$ , $λ \in R ∖ {0}$ , and $g (z) = λ ∣ z ∣^{m - 1} z$ . Write $H_{0}^{1} (Ω)$ for the closure of $C_{c}^{\infty} (Ω; C)$ in the norm $∥ w ∥_{H^{1}} = (∥ w ∥_{2}^{2} + \sum_{j = 1}^{n} ∥ D_{j} w ∥_{2}^{2})^{1 / 2}$ , where $D_{j}$ denotes the weak derivative. The $L^{\infty}$ norm is the essential supremum, and $∣ \nabla w ∣ = (\sum_{j} ∣ D_{j} w ∣^{2})^{1 / 2}$ .

There is a constant $C$ depending only on $m$ and $∣ λ ∣$ such that, for $u, v \in H_{0}^{1} (Ω) \cap L^{\infty} (Ω)$ , $g (v) - g (u) \in H_{0}^{1} (Ω)$ and almost everywhere

$∣ g (v) - g (u) ∣ \leq C (∣ v ∣^{m - 1} + ∣ u ∣^{m - 1}) ∣ v - u ∣,$

$∣ \nabla (g (v) - g (u)) ∣ \leq C (∣ v ∣^{m - 1} + ∣ u ∣^{m - 1}) ∣ \nabla (v - u) ∣ + C (∣ v ∣^{m - 2} + ∣ u ∣^{m - 2}) (∣ \nabla v ∣ + ∣ \nabla u ∣) ∣ v - u ∣ .$

Consequently, with the displayed norm on $H_{0}^{1} (Ω)$ ,

$∥ g (v) - g (u) ∥_{H_{0}^{1}} \leq C (∥ v ∥_{\infty}^{m - 1} + ∥ u ∥_{\infty}^{m - 1}) ∥ v - u ∥_{H_{0}^{1}} + C (∥ v ∥_{\infty}^{m - 2} + ∥ u ∥_{\infty}^{m - 2}) (∥ v ∥_{H_{0}^{1}} + ∥ u ∥_{H_{0}^{1}}) ∥ v - u ∥_{\infty} .$

Statement references

Power nonlinearity preserves the zero-boundary energy space

Complex polynomial weak derivatives and zero boundary preservation

Complex integer-order Sobolev notation

The Axiom of Choice

Facts & Assumptions

Given: The hypotheses and notation in the Statement.

[F1]

The complex Sobolev norm, weak derivatives, essential-supremum convention and zero-boundary closure are as in Complex integer-order Sobolev notation. The assumed Axiom of Choice is recorded in The Axiom of Choice.

[F2]

Under these hypotheses Power nonlinearity preserves the zero-boundary energy space gives $g (u), g (v) \in H_{0}^{1} (Ω)$ and the weak gradient formula for $g$ .

[F3]

The polynomial energy rule Complex polynomial weak derivatives and zero boundary preservation applies under Choice on open sets to bounded $H^{1}$ functions. It gives $D_{j} g (w) = λ (a w^{a - 1} {\overset{ˉ}{w}}^{b} D_{j} w + b w^{a} {\overset{ˉ}{w}}^{b - 1} \overline{D_{j} w})$ , where $a = (m + 1) / 2$ and $b = (m - 1) / 2$ , and supplies the corresponding $H^{1}$ difference estimate.

Proof

Given: The hypotheses and notation in the Statement.

[F1] The complex Sobolev norm, weak derivatives, essential-supremum convention and zero-boundary closure are as in Complex integer-order Sobolev notation. The assumed Axiom of Choice is recorded in The Axiom of Choice.

[F2] Under these hypotheses Power nonlinearity preserves the zero-boundary energy space gives $g (u), g (v) \in H_{0}^{1} (Ω)$ and the weak gradient formula for $g$ .

[F3] The polynomial energy rule Complex polynomial weak derivatives and zero boundary preservation applies under Choice on open sets to bounded $H^{1}$ functions. It gives $D_{j} g (w) = λ (a w^{a - 1} {\overset{ˉ}{w}}^{b} D_{j} w + b w^{a} {\overset{ˉ}{w}}^{b - 1} \overline{D_{j} w})$ , where $a = (m + 1) / 2$ and $b = (m - 1) / 2$ , and supplies the corresponding $H^{1}$ difference estimate.

Given: The hypotheses and functions in the Statement.

Goal: Establish the two pointwise inequalities and the energy-norm inequality.

1.1F1F2givenalgebra

Put $h = v - u$ , $a = (m + 1) / 2$ and $b = (m - 1) / 2$ . Since $g (z) = λ z^{a} {\overset{ˉ}{z}}^{b}$ , telescope the $m$ factors in $v^{a} {\overset{ˉ}{v}}^{b} - u^{a} {\overset{ˉ}{u}}^{b}$ . Each summand has one factor $h$ or $\overset{ˉ}{h}$ and $m - 1$ factors chosen from $u, \overset{ˉ}{u}, v, \overset{ˉ}{v}$ . Hence $∣ g (v) - g (u) ∣ \leq m ∣ λ ∣ (∣ u ∣ + ∣ v ∣)^{m - 1} ∣ h ∣$ . The elementary inequality $(x + y)^{r} \leq 2^{r - 1} (x^{r} + y^{r})$ for $x, y \geq 0$ and integer $r \geq 1$ gives the first displayed bound. By [F2], both nonlinear terms lie in $H_{0}^{1}$ , so their difference does as well.

2.1F3step 1.1algebra

In the derivative formula of [F3], the two coefficient polynomials have degree $m - 1$ in $w$ and $\overset{ˉ}{w}$ . Their absolute values are bounded by a constant times $∣ w ∣^{m - 1}$ ; telescoping their factors shows that each coefficient difference between $v$ and $u$ is bounded by a constant times $(∣ v ∣^{m - 2} + ∣ u ∣^{m - 2}) ∣ h ∣$ . Subtract the two formulas, writing each derivative of $v$ as the corresponding derivative of $u$ plus that of $h$ . The triangle inequality in $C^{n}$ then yields the stated gradient inequality, after increasing $C$ ; conjugation does not change the vector norm.

3.1F1F3step 1.1step 2.1algebra∎

Take $L^{2}$ norms in the value and gradient inequalities, bound the coefficients and $h$ by their essential suprema, and use the triangle inequality for the value-and-gradient vector defining the $H^{1}$ norm. This gives the displayed energy bound, with a possibly larger $C$ depending only on $m$ and $∣ λ ∣$ . The same norm inequality is also supplied directly by [F3]; [F1] identifies the norm on the zero-boundary subspace.

Source notes

The article is cited for attribution of the pointwise and energy difference bounds used in Proposition 3.9. The argument above uses the separately supplied weak polynomial rule and zero-boundary preservation, not the article's claim as a premise.

Depends on

Used by · 1

Cubic and quintic energy-space nonlinear time estimates

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Nonlinear Schrödinger equation in an exterior domain, v2 · PDF pages 15–16, proof of Proposition 3.9