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Lifting zero-order observability to the order-two endpoint

PDF p. 12: σ=2 part of Proposition 3.5 proof

Statement

Assume the Axiom of Choice. Let $n \geq 2$ , let $Ω = R^{n} ∖ Θ$ be the smooth connected exterior of a nonempty compact obstacle, and let $φ$ be the real control cutoff of Control region and nested real cutoffs. Let $Δ_{D}$ be the Dirichlet Laplacian, $A = I - Δ_{D}$ , $H_{D}^{s} (Ω)$ the associated spectral scale with $∥ u ∥_{H_{D}^{2}} = ∥ A u ∥_{L^{2}}$ , and $W (t) = e^{i t Δ_{D}}$ , as in Dirichlet Laplacian and positive shift and Dirichlet spectral Sobolev scale and duality. Fix $0 < T < \infty$ .

Suppose there is $C_{0} > 0$ such that every $f \in L^{2} (Ω)$ satisfies the zero-order observability inequality $∥ f ∥_{L^{2}}^{2} \leq C_{0} \int_{0}^{T} ∥ φ W (t) f ∥_{L^{2}}^{2} d t .$ Then there is $C > 0$ , depending on $T$ , $C_{0}$ and the fixed cutoff and scale constants, such that every $v_{0} \in H_{D}^{2} (Ω)$ , with $v (t) = W (t) v_{0}$ , satisfies $∥ v_{0} ∥_{H_{D}^{2}}^{2} \leq C \int_{0}^{T} ∥ φ v (t) ∥_{H_{D}^{2}}^{2} d t .$ In particular, before absorption there are constants $C_{1}, C_{2} > 0$ such that $∥ v_{0} ∥_{H_{D}^{2}}^{2} \leq C_{1} \int_{0}^{T} ∥ φ v (t) ∥_{H_{D}^{2}}^{2} d t + C_{2} T ∥ v_{0} ∥_{H_{D}^{1}}^{2} .$ The zero-order inequality is an explicit hypothesis of this conditional lifting statement.

Facts & Assumptions

[A1]

The zero-order observability inequality in the Statement holds for every $L^{2}$ initial datum, with the same constant $C_{0}$ .

[F1]

The cutoff $φ$ is smooth and real, vanishes near the obstacle and has bounded derivatives; $Δ_{D}$ has domain $H_{D}^{2} = D (A)$ , and $∥ u ∥_{H_{D}^{2}} = ∥ A u ∥_{2}$ . Control region and nested real cutoffs Dirichlet Laplacian and positive shift Dirichlet spectral Sobolev scale and duality

[F2]

Under Choice, $C_{φ} = [Δ, φ]$ extends boundedly from $H_{D}^{1}$ to $L^{2}$ . For $u \in D (A)$ , $φ u \in D (A)$ and $A (φ u) = φ A u - C_{φ} u$ . The domain and identity follow from the integer commutator supplier by writing $u = A^{- 1} g$ , $g \in L^{2}$ , in its resolvent identity. Integer resolvent commutator and localized first-order bounds The Axiom of Choice

[F3]

Under Choice, $W (t)$ is a strongly continuous unitary group on each scale space. It commutes with the scale powers; in particular $A W (t) u = W (t) A u$ for $u \in D (A)$ . Multiplication by $φ$ is bounded on $H_{D}^{2}$ . Scale isometries and free Schrödinger evolution Cutoff multipliers through order two and by duality The Axiom of Choice

[F4]

For each $ε > 0$ , $∥ u ∥_{H_{D}^{1}}^{2} \leq ε ∥ u ∥_{H_{D}^{2}}^{2} + C_{ε} ∥ u ∥_{2}^{2}$ for $u \in H_{D}^{2}$ . Also $∥ u ∥_{2} \leq ∥ u ∥_{H_{D}^{2}}$ . Spectral interpolation and lower-order absorption

[F5]

The time integrals are integrals of squared norms of strongly measurable scale-valued functions. Continuous and mixed time-space function spaces

Proof

[A1] The zero-order observability inequality in the Statement holds for every $L^{2}$ initial datum, with the same constant $C_{0}$ .

[F1] The cutoff $φ$ is smooth and real, vanishes near the obstacle and has bounded derivatives; $Δ_{D}$ has domain $H_{D}^{2} = D (A)$ , and $∥ u ∥_{H_{D}^{2}} = ∥ A u ∥_{2}$ . Control region and nested real cutoffs Dirichlet Laplacian and positive shift Dirichlet spectral Sobolev scale and duality

[F2] Under Choice, $C_{φ} = [Δ, φ]$ extends boundedly from $H_{D}^{1}$ to $L^{2}$ . For $u \in D (A)$ , $φ u \in D (A)$ and $A (φ u) = φ A u - C_{φ} u$ . The domain and identity follow from the integer commutator supplier by writing $u = A^{- 1} g$ , $g \in L^{2}$ , in its resolvent identity. Integer resolvent commutator and localized first-order bounds The Axiom of Choice

[F3] Under Choice, $W (t)$ is a strongly continuous unitary group on each scale space. It commutes with the scale powers; in particular $A W (t) u = W (t) A u$ for $u \in D (A)$ . Multiplication by $φ$ is bounded on $H_{D}^{2}$ . Scale isometries and free Schrödinger evolution Cutoff multipliers through order two and by duality The Axiom of Choice

[F4] For each $ε > 0$ , $∥ u ∥_{H_{D}^{1}}^{2} \leq ε ∥ u ∥_{H_{D}^{2}}^{2} + C_{ε} ∥ u ∥_{2}^{2}$ for $u \in H_{D}^{2}$ . Also $∥ u ∥_{2} \leq ∥ u ∥_{H_{D}^{2}}$ . Spectral interpolation and lower-order absorption

[F5] The time integrals are integrals of squared norms of strongly measurable scale-valued functions. Continuous and mixed time-space function spaces

Given: The geometry, cutoff, scale, Choice assumption, $T$ , the zero-order observability hypothesis, and $v_{0} \in H_{D}^{2}$ in the Statement.

Goal: Prove the displayed preliminary estimate and absorb its lower-order term to obtain the order-two observability inequality.

1.1A1F1F2F3F5

Put $v (t) = W (t) v_{0}$ . By scale unitarity, $v (t) \in H_{D}^{2}$ , $∥ v (t) ∥_{H_{D}^{1}} = ∥ v_{0} ∥_{H_{D}^{1}}$ , and $A v (t) = W (t) A v_{0}$ . The cutoff multiplier and strong continuity give $φ v \in C ([0, T]; H_{D}^{2})$ , so its squared norm is integrable. Apply the zero-order hypothesis to $A v_{0} \in L^{2}$ and use the commutator identity to obtain $∥ v_{0} ∥_{H_{D}^{2}}^{2} \leq C_{0} \int_{0}^{T} (∥ φ v (t) ∥_{H_{D}^{2}} + K ∥ v_{0} ∥_{H_{D}^{1}})^{2} d t$ , where $K$ is a bound for $C_{φ} : H_{D}^{1} \to L^{2}$ . Consequently the preliminary estimate holds with, for example, $C_{1} = 2 C_{0}$ and $C_{2} = 2 C_{0} K^{2}$ , enlarged if necessary to be positive.

2.1A1F3F4step 1.1∎

Apply the absorption inequality to $v_{0}$ , and apply zero-order observability to $v_{0}$ itself. Since $∥ φ v (t) ∥_{2} \leq ∥ φ v (t) ∥_{H_{D}^{2}}$ , these give $∥ v_{0} ∥_{H_{D}^{1}}^{2} \leq ε ∥ v_{0} ∥_{H_{D}^{2}}^{2} + C_{ε} C_{0} \int_{0}^{T} ∥ φ v (t) ∥_{H_{D}^{2}}^{2} d t$ . Choose $ε > 0$ with $C_{2} T ε \leq 1 / 2$ , substitute this estimate into step 1.1, and move the resulting half of $∥ v_{0} ∥_{H_{D}^{2}}^{2}$ to the left. The remaining finite constant proves the order-two inequality.

Source notes

The article is cited for attribution of the endpoint argument in the proof of Proposition 3.5 on PDF page 12. The proof above uses the stated conditional zero-order hypothesis and the independent mathematical suppliers; it does not assume the article's endpoint conclusion.

Depends on

Used by · 1

Proposition 3.5: full-scale free observability

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2