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Free HUM identity, coercivity and invertibility

PDF p. 12: pairing derivative at foot of page; PDF p. 13: HUM identity and Lax–Milgram

Statement

Assume the Axiom of Choice and hypothesis (H): $n \geq 2$ , the nonempty compact obstacle has smooth boundary and connected exterior $Ω$ , and its generalized reflected rays leave every compact set in finite time. Let $T > 0$ , $s \in [- 2, 2]$ , $σ = - s$ , and let $φ$ be the real control cutoff of Control region and nested real cutoffs. Write $H_{D}^{r}$ for the complex Dirichlet spectral scale, $W (t) = e^{i t Δ_{D}}$ , and $⟨ \cdot, \cdot ⟩_{s, σ}$ for its pivot pairing, linear in the first argument. Let $S : H_{D}^{σ} \to H_{D}^{s}$ be the HUM map of Scale Riesz map and free HUM operator.

For $v_{0}, w_{0} \in H_{D}^{σ}$ , put $v (t) = W (t) v_{0}$ , $w (t) = W (t) w_{0}$ , $h (t) = - i Λ_{s}^{- 1} (φ v (t))$ , and let $z$ be the backward mild solution with terminal value zero used in the definition of $S$ . Then $\frac{d}{d t} ⟨ z (t), w (t) ⟩_{s, σ} = - i ⟨ φ h (t), w (t) ⟩_{s, σ}$ for $0 < t < T$ , and $⟨ S v_{0}, w_{0} ⟩_{s, σ} = \int_{0}^{T} (φ v (t), φ w (t))_{H_{D}^{σ}} d t .$ In particular, for the observability constant $C > 0$ from Proposition 3.5: full-scale free observability, $⟨ S v_{0}, v_{0} ⟩_{s, σ} = \int_{0}^{T} ∥ φ W (t) v_{0} ∥_{H_{D}^{σ}}^{2} d t \geq C^{- 1} ∥ v_{0} ∥_{H_{D}^{σ}}^{2} .$ The bounded complex-linear map $S : H_{D}^{σ} \to H_{D}^{s}$ is an isomorphism with bounded inverse.

Facts & Assumptions

[F1]

The scale is a complex Hilbert scale with a continuous pivot pairing; $Λ_{s}^{- 1} = A^{- s} : H_{D}^{σ} \to H_{D}^{s}$ is the inverse scale Riesz map, characterized by $⟨ Λ_{s}^{- 1} a, b ⟩_{s, σ} = (a, b)_{H_{D}^{σ}}$ for $a, b \in H_{D}^{σ}$ . The HUM construction makes $S$ bounded and complex-linear. Dirichlet spectral Sobolev scale and duality Scale Riesz map and free HUM operator

[F2]

$W (t)$ is a strongly continuous unitary group on each scale level and preserves the pivot pairing: $⟨ W (t) f, W (t) g ⟩_{s, σ} = ⟨ f, g ⟩_{s, σ}$ . The real cutoff acts boundedly on both relevant levels and is symmetric for the pivot pairing: $⟨ φ f, g ⟩_{s, σ} = ⟨ f, φ g ⟩_{s, σ}$ . Scale isometries and free Schrödinger evolution Cutoff multipliers through order two and by duality Control region and nested real cutoffs

[F3]

For continuous $h : [0, T] \to H_{D}^{s}$ , the backward mild solution with $z (T) = 0$ satisfies $z (t) = i \int_{t}^{T} W (t - r) φ h (r) d r$ as a Bochner integral. Bounded linear functionals commute with this integral. Linear forced well-posedness by Duhamel formula Bounded linear maps commute with Bochner integration

[F4]

Under the stated geometry and Choice, observability gives $∥ x ∥_{H_{D}^{σ}}^{2} \leq C \int_{0}^{T} ∥ φ W (t) x ∥_{H_{D}^{σ}}^{2} d t$ for every $x \in H_{D}^{σ}$ . A bounded Hilbert-space observation with a positive lower bound has an invertible Gramian. Proposition 3.5: full-scale free observability Observation lower bounds give Hilbert control right inverses The Axiom of Choice

[F5]

The notation $L^{2} (0, T; X)$ means the almost-everywhere classes of strongly measurable maps on the Lebesgue interval with norm $(\int_{0}^{T} ∥ u (t) ∥_{X}^{2} d t)^{1 / 2}$ , with pointwise vector operations. Strong measurability uses finite-range measurable simple approximants. This construction is a normed space for real or complex Banach $X$ . Continuous and mixed time-space function spaces

[F6]

Restricting the scalars of a complex normed space to the reals preserves its set, norm and metric. For a real Banach space $X$ and $0 < T < \infty$ , the Bochner space $L^{2} (0, T; X)$ is complete under Countable Choice. The assumed Axiom of Choice supplies Countable Choice for this completeness theorem and the Gramian lemma. Realification preserves a complex norm and its metric Bochner Lp is complete for a Banach value space AC implies DC implies countable choice

[F7]

In a complex Hilbert space, Cauchy–Schwarz bounds the modulus of the inner product by the product of the norms. Scalar Hölder with exponents $(2, 2)$ bounds the integral of that norm product. Pointwise limits of measurable real functions are measurable, applied componentwise to complex functions. Complex Lebesgue integration is defined through real and imaginary parts, is linear on $L^{1}$ , and respects almost-everywhere equality. A complex inner-product space complete in its induced norm is a complex Hilbert space. Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs Holder's inequality for integrals, including the endpoint cases Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable Integrable real and complex functions, and their integrals The Lebesgue integral is linear on L¹(μ) Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree Hilbert space

[F8]

Assuming Countable Choice, if $X$ is a real or complex Banach space and $f : [a, b] \to X$ is continuous with $a < b$ , then $G (t) = \int_{a}^{t} f (r) d r$ is differentiable on $(a, b)$ with $G^{'} (t) = f (t)$ , using real differentiation. The assumed Axiom of Choice supplies Countable Choice. Fundamental theorem of calculus for Banach-valued continuous curves AC implies DC implies countable choice

Proof

[F1] The scale is a complex Hilbert scale with a continuous pivot pairing; $Λ_{s}^{- 1} = A^{- s} : H_{D}^{σ} \to H_{D}^{s}$ is the inverse scale Riesz map, characterized by $⟨ Λ_{s}^{- 1} a, b ⟩_{s, σ} = (a, b)_{H_{D}^{σ}}$ for $a, b \in H_{D}^{σ}$ . The HUM construction makes $S$ bounded and complex-linear. Dirichlet spectral Sobolev scale and duality Scale Riesz map and free HUM operator

[F2] $W (t)$ is a strongly continuous unitary group on each scale level and preserves the pivot pairing: $⟨ W (t) f, W (t) g ⟩_{s, σ} = ⟨ f, g ⟩_{s, σ}$ . The real cutoff acts boundedly on both relevant levels and is symmetric for the pivot pairing: $⟨ φ f, g ⟩_{s, σ} = ⟨ f, φ g ⟩_{s, σ}$ . Scale isometries and free Schrödinger evolution Cutoff multipliers through order two and by duality Control region and nested real cutoffs

[F3] For continuous $h : [0, T] \to H_{D}^{s}$ , the backward mild solution with $z (T) = 0$ satisfies $z (t) = i \int_{t}^{T} W (t - r) φ h (r) d r$ as a Bochner integral. Bounded linear functionals commute with this integral. Linear forced well-posedness by Duhamel formula Bounded linear maps commute with Bochner integration

[F4] Under the stated geometry and Choice, observability gives $∥ x ∥_{H_{D}^{σ}}^{2} \leq C \int_{0}^{T} ∥ φ W (t) x ∥_{H_{D}^{σ}}^{2} d t$ for every $x \in H_{D}^{σ}$ . A bounded Hilbert-space observation with a positive lower bound has an invertible Gramian. Proposition 3.5: full-scale free observability Observation lower bounds give Hilbert control right inverses The Axiom of Choice

[F5] The notation $L^{2} (0, T; X)$ means the almost-everywhere classes of strongly measurable maps on the Lebesgue interval with norm $(\int_{0}^{T} ∥ u (t) ∥_{X}^{2} d t)^{1 / 2}$ , with pointwise vector operations. Strong measurability uses finite-range measurable simple approximants. This construction is a normed space for real or complex Banach $X$ . Continuous and mixed time-space function spaces

[F6] Restricting the scalars of a complex normed space to the reals preserves its set, norm and metric. For a real Banach space $X$ and $0 < T < \infty$ , the Bochner space $L^{2} (0, T; X)$ is complete under Countable Choice. The assumed Axiom of Choice supplies Countable Choice for this completeness theorem and the Gramian lemma. Realification preserves a complex norm and its metric Bochner Lp is complete for a Banach value space AC implies DC implies countable choice

[F7] In a complex Hilbert space, Cauchy–Schwarz bounds the modulus of the inner product by the product of the norms. Scalar Hölder with exponents $(2, 2)$ bounds the integral of that norm product. Pointwise limits of measurable real functions are measurable, applied componentwise to complex functions. Complex Lebesgue integration is defined through real and imaginary parts, is linear on $L^{1}$ , and respects almost-everywhere equality. A complex inner-product space complete in its induced norm is a complex Hilbert space. Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs Holder's inequality for integrals, including the endpoint cases Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable Integrable real and complex functions, and their integrals The Lebesgue integral is linear on L¹(μ) Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree Hilbert space

[F8] Assuming Countable Choice, if $X$ is a real or complex Banach space and $f : [a, b] \to X$ is continuous with $a < b$ , then $G (t) = \int_{a}^{t} f (r) d r$ is differentiable on $(a, b)$ with $G^{'} (t) = f (t)$ , using real differentiation. The assumed Axiom of Choice supplies Countable Choice. Fundamental theorem of calculus for Banach-valued continuous curves AC implies DC implies countable choice

Given: The geometry, Choice, scale, cutoff, time and data in the Statement.

Goal: Establish the pairing derivative, HUM identity, coercivity and bounded invertibility.

1.1givenF1F2F3F6F8

The free paths $v, w$ are continuous in $H_{D}^{σ}$ . Boundedness of the cutoff and inverse Riesz map makes $h$ continuous in $H_{D}^{s}$ . By the backward formula and invariance of the pivot pairing, for every $t \in [0, T]$ , $⟨ z (t), w (t) ⟩_{s, σ} = i \int_{t}^{T} ⟨ φ h (r), w (r) ⟩_{s, σ} d r .$ The scalar integrand $f (r) = ⟨ φ h (r), w (r) ⟩_{s, σ}$ is continuous. Apply F8 with $X = C$ , $a = 0$ , $b = T$ and this $f$ ; Countable Choice is available by F6. Writing $i \int_{t}^{T} f (r) d r = i (G (T) - G (t))$ gives derivative $- i f (t)$ on $(0, T)$ , which is the stated pairing derivative without requiring a classical boundary trace.

1.2givenF1F5F6

Put $E = H_{D}^{σ}$ and let $U = L^{2} (0, T; E)$ have the precise meaning in F5. The underlying real space $E_{R}$ is Banach: $E$ is Hilbert by F1 and restriction of scalars preserves its complete norm metric by F6. Strong measurability into $E$ and into $E_{R}$ is the same condition, since the finite-range simple functions, their measurability, and norm convergence are unchanged. Thus $L^{2} (0, T; E_{R})$ and $U$ have exactly the same classes and norm metric. Apply the real Bochner completeness theorem with $X = E_{R}$ , $p = 2$ and the given finite $T > 0$ . Countable Choice is available by F6, so $U$ is complete. Its complex vector operations and norm are those supplied by F5; no separability hypothesis on $E$ is required.

2.1F1F5F7step 1.2

For $u, v \in U$ , define $(u, v)_{U} = \int_{0}^{T} (u (t), v (t))_{E} d t$ . Choose simple approximants for $u$ and $v$ ; their scalar inner products are measurable simple functions. Cauchy–Schwarz gives $∣ (a, b)_{E} - (u, v)_{E} ∣ \leq ∥ a - u ∥_{E} ∥ b ∥_{E} + ∥ u ∥_{E} ∥ b - v ∥_{E}$ , so the inner products of the approximants converge almost everywhere to $(u (t), v (t))_{E}$ . After setting representatives to zero on the exceptional null sets, componentwise measurable limits give a measurable scalar integrand. Pointwise Cauchy–Schwarz and scalar Hölder give $\int_{0}^{T} ∣ (u (t), v (t))_{E} ∣ d t \leq ∥ u ∥_{U} ∥ v ∥_{U} < \infty$ . Almost-everywhere invariance of the integral makes the pairing independent of representatives. Linearity of the complex integral gives linearity in the first variable and conjugate-linearity in the second; the componentwise definition also gives $\int \overline{f} = \overline{\int f}$ , so conjugate symmetry is inherited from $E$ . Finally $(u, u)_{U} = \int_{0}^{T} ∥ u (t) ∥_{E}^{2} d t = ∥ u ∥_{U}^{2}$ , which is positive definite on the quotient by F5. Hence this is a complex inner product with the stated norm. Completeness from step 1.2 makes $U$ a complex Hilbert space.

2.2F1F2F4step 1.1

At $t = 0$ , substitute $h = - i Λ_{s}^{- 1} (φ v)$ into the integral in step 1.1. Since the pairing is linear in its first argument, $i (- i) = 1$ . Cutoff symmetry and the Riesz identity give $i ⟨ φ h, w ⟩_{s, σ} = (φ v, φ w)_{H_{D}^{σ}}$ . As $z (0) = S v_{0}$ , integration proves the displayed bilinear HUM identity. Setting $w_{0} = v_{0}$ and applying observability gives the stated coercive estimate.

3.1F1F2F4F5step 2.1

Define the observation $O : E \to U$ by $(O x) (t) = φ W (t) x$ . This path is continuous in $E$ . Its step functions sampling the left endpoints of equal time subdivisions converge in norm at each interior time, so it is strongly measurable in the sense of F5. Unitarity and cutoff boundedness give $∥ O x ∥_{U} \leq \sqrt{T} ∥ φ ∥ ∥ x ∥_{E}$ , and the map is complex-linear. Observability gives $∥ O x ∥_{U} \geq C^{- 1 / 2} ∥ x ∥_{E}$ . Since both $E$ and $U$ are complex Hilbert spaces, the Gramian lemma now applies and makes $Q = O^{*} O$ a bounded isomorphism of $E$ .

4.1F1F4step 2.1step 3.1step 2.2∎

The Hilbert adjoint identity gives $(Q v_{0}, w_{0})_{E} = (O v_{0}, O w_{0})_{U} = \int_{0}^{T} (φ v (t), φ w (t))_{E} d t$ , using the integrated inner product constructed in step 2.1. Comparing with step 2.2 and the Riesz identity yields $S v_{0} = Λ_{s}^{- 1} Q v_{0}$ : both sides have the same pivot pairing with every $w_{0}$ , and the pairing is nondegenerate. Both factors are bounded isomorphisms, so $S$ is a bounded complex-linear isomorphism with inverse $Q^{- 1} Λ_{s}$ .

Source notes

The original article is cited for attribution of the pairing calculation and HUM argument on PDF pages 12–13. The proof above uses the stated scale, Duhamel, observability and Gramian suppliers. The complex Bochner Hilbert structure is constructed locally in steps 1.2–2.1 from the existing time-space definition and the independently supplied real Bochner completeness and scalar integration results; it is not imported from the article.

Depends on

Used by · 1

Theorem 3.6: bounded continuous control operator

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2