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Global Strichartz estimate with derivative loss

PDF p. 7: Proposition 2.3; (2.5)

Statement

Assume the Axiom of Choice. Let $n \geq 2$ and let $Ω = R^{n} ∖ Θ$ be the connected exterior of a nonempty compact smooth non-trapping obstacle. Let $Δ_{D}$ be the Dirichlet Laplacian, $H_{D}^{r} (Ω)$ its spectral Sobolev scale, and $W (t) = e^{i t Δ_{D}}$ . Fix $T > 0$ , $0 \leq s \leq 1$ , and exponents $2 < p \leq \infty$ , $2 \leq q < \infty$ satisfying $2 / p + n / q = n / 2$ , with $1 / \infty = 0$ . For $0 < s < 1$ use the complex interpolation convention $W^{s, q} (Ω) = [L^{q} (Ω), W^{1, q} (Ω)]_{s}$ , and set $W^{0, q} = L^{q}$ . Time-space norms are Bochner norms.

There is a constant $C$ , depending on the fixed domain, exponents, $s$ and $T$ , such that for every $u_{0} \in H_{D}^{s + 1 / p} (Ω)$ , $∥ W (t) u_{0} ∥_{L^{p} (0, T; W^{s, q} (Ω))} \leq C ∥ u_{0} ∥_{H_{D}^{s + 1 / p} (Ω)} .$ This is a whole-domain estimate with a loss of $1 / p$ derivatives.

Statement references

Exterior and truncated obstacle domains

Non-trapping obstacle

Dirichlet spectral Sobolev scale and duality

Continuous and mixed time-space function spaces

An explicit common interpolation model for the fractional Strichartz targets

Integer-order Sobolev spaces and their norms

The Axiom of Choice

Facts & Assumptions

[F1]

The geometry definition gives a radius $r > 0$ with $Θ \subset B_{r} (0)$ , and the non-trapping condition is the stated generalized-ray escape hypothesis. Exterior and truncated obstacle domains Non-trapping obstacle

[F2]

The Dirichlet scale defines $H_{D}^{r}$ and the group $W (t)$ ; the time-space definition specifies the Bochner norms. Dirichlet spectral Sobolev scale and duality Continuous and mixed time-space function spaces

[F3]

The integer endpoint $W^{1, q} (Ω)$ is the complex weak-derivative Sobolev space with the norm of the integer Sobolev definition, applied with $k = 1$ , exponent $q$ and scalar field $C$ ; its countable-choice hypothesis follows from the assumed AC. The fractional target is the complex interpolation space in the statement, with its endpoint conventions. Integer-order Sobolev spaces and their norms An explicit common interpolation model for the fractional Strichartz targets The Axiom of Choice

[F4]

Given a smooth compactly supported cutoff equal to one near the obstacle, the completed exterior Strichartz proposition supplies the displayed whole-domain estimate for precisely these exponents and this target convention, including $p = \infty$ . Exterior Strichartz estimates with precise loss

[F5]

For $0 < r < R$ , a smooth function supported in $B_{R} (0)$ and equal to one on ${\overline{B}}_{r} (0)$ exists. A smooth bump between concentric Euclidean balls

[F6]

The Axiom of Choice is assumed, as required by the scale and interpolation contracts. The Axiom of Choice

[F7]

In $R^{n}$ with its Euclidean metric, a subset is compact if and only if it is closed and bounded. Thus the compact obstacle is bounded, and a closed support contained in a bounded ball is compact. Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line

Proof

[F1] The geometry definition gives a radius $r > 0$ with $Θ \subset B_{r} (0)$ , and the non-trapping condition is the stated generalized-ray escape hypothesis. Exterior and truncated obstacle domains Non-trapping obstacle

[F2] The Dirichlet scale defines $H_{D}^{r}$ and the group $W (t)$ ; the time-space definition specifies the Bochner norms. Dirichlet spectral Sobolev scale and duality Continuous and mixed time-space function spaces

[F3] The integer endpoint $W^{1, q} (Ω)$ is the complex weak-derivative Sobolev space with the norm of the integer Sobolev definition, applied with $k = 1$ , exponent $q$ and scalar field $C$ ; its countable-choice hypothesis follows from the assumed AC. The fractional target is the complex interpolation space in the statement, with its endpoint conventions. Integer-order Sobolev spaces and their norms An explicit common interpolation model for the fractional Strichartz targets The Axiom of Choice

[F4] Given a smooth compactly supported cutoff equal to one near the obstacle, the completed exterior Strichartz proposition supplies the displayed whole-domain estimate for precisely these exponents and this target convention, including $p = \infty$ . Exterior Strichartz estimates with precise loss

[F5] For $0 < r < R$ , a smooth function supported in $B_{R} (0)$ and equal to one on ${\overline{B}}_{r} (0)$ exists. A smooth bump between concentric Euclidean balls

[F6] The Axiom of Choice is assumed, as required by the scale and interpolation contracts. The Axiom of Choice

[F7] In $R^{n}$ with its Euclidean metric, a subset is compact if and only if it is closed and bounded. Thus the compact obstacle is bounded, and a closed support contained in a bounded ball is compact. Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line

Given: The obstacle, exponents, time, initial datum and Choice assumption in the statement.

Goal: Establish the displayed whole-domain estimate.

1.1F1F5F7givenchoose

The compact obstacle is bounded by F7, so choose $r > 0$ with $Θ \subset B_{r} (0)$ and choose $R > r$ . The smooth-bump lemma gives a smooth function $χ$ equal to one on ${\overline{B}}_{r} (0)$ with $supp χ \subset B_{R} (0)$ . Its support is the closure of its nonzero set, hence is closed; it is bounded by this containment and therefore compact by F7. Consequently $χ \in C_{c}^{\infty} (R^{n})$ and equals one on the open neighborhood $B_{r} (0)$ of $Θ$ . Thus $χ$ meets the cutoff hypothesis of the completed exterior estimate.

2.1F1F2F3F4F6step 1.1given∎

The domain, non-trapping hypothesis, Choice assumption and exponents meet the remaining hypotheses of that estimate. Its whole-domain conclusion applies to $u_{0}$ and, with the scale, group, Bochner norms and fractional target identified by the stated conventions, is exactly the displayed inequality. The constant may depend on the chosen cutoff, which is fixed once the domain is fixed, as well as on the other displayed data. This proves the assertion.

Source notes

The original article is cited for attribution to Proposition 2.3, equation (2.5); it is not used as proof here. The completed exterior estimate records its independent Burq–Gérard–Tzvetkov inputs and their reading limits. The explicitly chosen complex interpolation model does not assert an identification with Slobodeckij spaces for $q \neq 2$ .

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2