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Power nonlinearity preserves the zero-boundary energy space

PDF p. 15: Proposition 3.9 proof (iii), polynomial identity and gradient formula

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 $g (z) = λ ∣ z ∣^{m - 1} z$ . Let $H_{0}^{1} (Ω)$ be the closure of $C_{c}^{\infty} (Ω; C)$ in the complex weak-derivative norm $∥ u ∥_{H^{1}}^{2} = ∥ u ∥_{2}^{2} + \sum_{j = 1}^{n} ∥ D_{j} u ∥_{2}^{2}$ . The $L^{\infty}$ norm is the essential supremum. For $v \in H_{0}^{1} (Ω) \cap L^{\infty} (Ω)$ , $g (v) \in H_{0}^{1} (Ω)$ and, almost everywhere,

$\nabla g (v) = λ (\frac{m + 1}{2} ∣ v ∣^{m - 1} \nabla v + \frac{m - 1}{2} ∣ v ∣^{m - 3} v^{2} \nabla \overline{v}) .$

With the displayed $H^{1}$ norm on $H_{0}^{1} (Ω)$ ,

$∥ g (v) ∥_{H_{0}^{1}} \leq m ∣ λ ∣ ∥ v ∥_{L^{\infty}}^{m - 1} ∥ v ∥_{H_{0}^{1}} .$

This is also the norm of $v$ in the spectral space $H_{D}^{1} (Ω)$ associated with $I - Δ_{D}$ .

Statement references

Linear and nonlinear internally controlled exterior systems

Complex integer-order Sobolev notation

Dirichlet spectral Sobolev scale and duality

The Axiom of Choice

Facts & Assumptions

Given: The hypotheses and notation in the Statement.

[F1]

The complex polynomial energy rule Complex polynomial weak derivatives and zero boundary preservation applies under Choice on any open subset of Euclidean space. For bounded $H^{1}$ functions it supplies the weak derivative formula for $λ v^{a} {\overline{v}}^{b}$ , with $a = (m + 1) / 2$ and $b = (m - 1) / 2$ , and it preserves $H_{0}^{1}$ .

[F2]

The weak Sobolev norm and the zero-boundary closure are as in Complex integer-order Sobolev notation. The exterior geometry and the nonlinear function $g$ are those of Linear and nonlinear internally controlled exterior systems, and the assumed Choice is recorded by The Axiom of Choice.

[F3]

The Dirichlet scale Dirichlet spectral Sobolev scale and duality uses $H_{D}^{1} = D ((I - Δ_{D})^{1 / 2})$ . The exterior Dirichlet realization Exterior Dirichlet realization, H² domain and form domain identifies this space with $H_{0}^{1} (Ω)$ and its spectral norm squared with $∥ u ∥_{2}^{2} + \sum_{j} ∥ D_{j} u ∥_{2}^{2}$ .

Proof

Given: The hypotheses and notation in the Statement.

[F1] The complex polynomial energy rule Complex polynomial weak derivatives and zero boundary preservation applies under Choice on any open subset of Euclidean space. For bounded $H^{1}$ functions it supplies the weak derivative formula for $λ v^{a} {\overline{v}}^{b}$ , with $a = (m + 1) / 2$ and $b = (m - 1) / 2$ , and it preserves $H_{0}^{1}$ .

[F2] The weak Sobolev norm and the zero-boundary closure are as in Complex integer-order Sobolev notation. The exterior geometry and the nonlinear function $g$ are those of Linear and nonlinear internally controlled exterior systems, and the assumed Choice is recorded by The Axiom of Choice.

[F3] The Dirichlet scale Dirichlet spectral Sobolev scale and duality uses $H_{D}^{1} = D ((I - Δ_{D})^{1 / 2})$ . The exterior Dirichlet realization Exterior Dirichlet realization, H² domain and form domain identifies this space with $H_{0}^{1} (Ω)$ and its spectral norm squared with $∥ u ∥_{2}^{2} + \sum_{j} ∥ D_{j} u ∥_{2}^{2}$ .

Given: The hypotheses and data in the Statement.

Goal: Prove zero-boundary preservation, the gradient formula, and the stated norm estimate.

1.1F1F2givenalgebra

Put $a = (m + 1) / 2$ and $b = (m - 1) / 2$ . These are positive integers, and $v^{a} {\overline{v}}^{b} = ∣ v ∣^{m - 1} v$ . The exterior is open, so the polynomial energy rule applies to $v$ and gives $g (v) \in H_{0}^{1} (Ω)$ and $D_{j} g (v) = λ (a v^{a - 1} {\overline{v}}^{b} D_{j} v + b v^{a} {\overline{v}}^{b - 1} \overline{D_{j} v})$ almost everywhere. Since $v^{a - 1} {\overline{v}}^{b} = ∣ v ∣^{m - 1}$ and $v^{a} {\overline{v}}^{b - 1} = ∣ v ∣^{m - 3} v^{2}$ , this is the displayed gradient formula; $D_{j} \overline{v} = \overline{D_{j} v}$ in the weak sense.

2.1F2F3step 1.1algebra∎

Set $M = ∥ v ∥_{L^{\infty}}$ . The formula in step 1.1 and $a + b = m$ give $∣ D_{j} g (v) ∣ \leq m ∣ λ ∣ M^{m - 1} ∣ D_{j} v ∣$ almost everywhere. Also $∣ g (v) ∣ \leq ∣ λ ∣ M^{m - 1} ∣ v ∣$ . Squaring, integrating and summing the value and derivative bounds yields $∥ g (v) ∥_{H_{0}^{1}} \leq m ∣ λ ∣ M^{m - 1} ∥ v ∥_{H_{0}^{1}}$ , including the case $M = 0$ . The form-norm identity in [F3] gives the final spectral-norm assertion.

Source notes

The article is cited for attribution of the polynomial identity, gradient formula and estimate in Proposition 3.9. The proof above uses the separately supplied polynomial energy rule and Dirichlet realization.

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2