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Proposition 3.9: energy-space local control

PDF p. 15: Proposition 3.9; PDF p. 16: proof concludes

Statement

Assume the Axiom of Choice and hypothesis (H): $n \in {2, 3}$ , $Θ \subset R^{n}$ is nonempty and compact with smooth boundary, its exterior $Ω = R^{n} ∖ Θ$ is connected, and every generalized reflected or glancing ray leaves each compact set in finite forward time. Let $T > 0$ , $m \in {3, 5}$ and $λ \in R ∖ {0}$ . Choose the control cutoff $φ$ of Control region and nested real cutoffs.

Let $H_{0}^{1} (Ω)$ be the closure of $C_{c}^{\infty} (Ω; C)$ in $H^{1} (Ω)$ , and let $X_{T} = C ([0, T]; H_{0}^{1} (Ω)) \cap L_{t}^{4} L_{x}^{\infty}$ , with compatible jointly measurable representatives and norm $∥ v ∥_{X_{T}} = \sup_{t \in [0, T]} ∥ v (t) ∥_{H^{1}} + ∥ v ∥_{L_{t}^{4} L_{x}^{\infty}}$ . The mixed norm is the $L^{4}$ norm in time of the spatial essential supremum. At $s = 1$ , this space satisfies the homogeneous, retarded-forcing and nonlinear estimates in the abstract perturbation hypotheses. After an equivalent constant rescaling of its norm if needed, it satisfies their norm-one inclusion convention as well.

Consequently, there is $δ > 0$ such that for every $v_{0}, v_{T} \in H_{0}^{1} (Ω)$ with $∥ v_{0} ∥_{H^{1}}, ∥ v_{T} ∥_{H^{1}} \leq δ$ , there exist $h \in C ([0, T]; H_{0}^{1} (Ω))$ and $v \in X_{T}$ solving $i \partial_{t} v + Δ_{D} v + λ ∣ v ∣^{m - 1} v = φ h$ in the integral sense, with $v (0) = v_{0}$ and $v (T) = v_{T}$ . The homogeneous Dirichlet condition is encoded by $Δ_{D}$ . Thus the internally controlled system is locally exactly controllable in $H_{0}^{1} (Ω)$ .

Statement references

Dirichlet Laplacian and positive shift

Mild formulation of the controlled power NLS

Exterior and truncated obstacle domains

Control region and nested real cutoffs

Linear and nonlinear internally controlled exterior systems

Energy control space with a time-integrable supremum norm

Abstract nonlinear control-space hypotheses

Complex integer-order Sobolev notation

Continuous and mixed time-space function spaces

The Axiom of Choice

Facts & Assumptions

[F1]

The stated geometry and ray convention are those of Exterior and truncated obstacle domains. The cutoff and controlled equation are specified by Control region and nested real cutoffs and Linear and nonlinear internally controlled exterior systems. The stated Sobolev closure and scalar mixed-norm conventions are those of Complex integer-order Sobolev notation and Continuous and mixed time-space function spaces. The Axiom of Choice The Dirichlet operator is the realization in Dirichlet Laplacian and positive shift.

[F2]

Energy control space with a time-integrable supremum norm supplies completeness of $X_{T}$ with the displayed norm. Low-order identifications of the Dirichlet scale identifies $H_{D}^{1} (Ω)$ and $H_{0}^{1} (Ω)$ as sets with equivalent norms.

[F3]

Under the stated geometry and Choice assumption, Free and retarded bounds in the energy control space gives the homogeneous evolution and retarded Bochner-forcing estimates in $X_{T}$ with $H_{0}^{1}$ norms. Cubic and quintic energy-space nonlinear time estimates gives $g (v) \in L^{1} (0, T; H_{0}^{1})$ and the required growth and difference estimates for $g (z) = λ ∣ z ∣^{m - 1} z$ ; its time factor is $T^{(5 - m) / 4}$ , including the value $1$ for $m = 5$ .

[F4]

Abstract nonlinear control-space hypotheses specifies the Banach path space, norm-one inclusion and three estimates used by Theorem 3.7: abstract small-data perturbation result. That theorem gives a continuous control and an integral solution joining sufficiently small endpoints in $H_{D}^{1}$ when those hypotheses hold. The integral-solution convention and its mild equivalence are those of Mild formulation of the controlled power NLS.

Proof

Given: The assumptions and notation of the Statement.

Goal: Verify the abstract hypotheses at $s = 1$ and obtain the stated local exact control.

1.1F2F4given

Put $H = H_{D}^{1} (Ω)$ . By F2, there are positive constants $a, b$ such that $a ∥ w ∥_{H^{1}} \leq ∥ w ∥_{H} \leq b ∥ w ∥_{H^{1}}$ for every $w \in H_{0}^{1}$ . Choose $c \geq \max {1, b}$ and equip the same vector space $X_{T}$ with $∥ v ∥_{\hat{X}} = c ∥ v ∥_{X_{T}}$ . This equivalent norm preserves completeness, and $\sup_{t} ∥ v (t) ∥_{H} \leq ∥ v ∥_{\hat{X}}$ . Hence the path-space and norm-one inclusion requirements in F4 hold.

2.1F2F3F4step 1.1

The homogeneous and retarded estimates of F3 apply to exactly this exterior, time interval and space. Replacing the $H_{0}^{1}$ norms by the equivalent $H$ norms and the $X_{T}$ norm by $∥ \cdot ∥_{\hat{X}}$ changes only their constants. The same equivalence carries strong measurability and Bochner integrability between the two descriptions of $H$ , so the forcing estimate has the required meaning. These are conditions (i) and (ii) of F4.

3.1F2F3F4step 1.1step 2.1

By F3, $g (v)$ is Bochner integrable in $H_{0}^{1}$ and satisfies the growth and difference bounds for every $u, v \in X_{T}$ . Norm equivalence and the rescaling in step 1.1 give the corresponding bounds with the $H$ and $\hat{X}$ norms. Since $T > 0$ is fixed, $T^{(5 - m) / 4}$ is a finite constant, including when $m = 5$ . Taking one sufficiently large constant for these and the two linear bounds verifies condition (iii) and the common-constant requirement of F4.

4.1F1F2F4step 1.1step 2.1step 3.1∎

Apply the theorem in F4 with $s = 1$ and the rescaled space. It yields a positive endpoint radius in the $H$ norm, a control continuous in $H$ , and an integral solution in $X_{T}$ with the prescribed endpoints. Choose $δ$ small enough that $b δ$ is at most that radius. The norm equivalence makes the control continuous in $H_{0}^{1}$ and places every endpoint pair in the theorem's range, proving the stated conclusion.

Source notes

The frozen article is cited for attribution of Proposition 3.9. The application above uses the separately supplied linear and nonlinear estimates and the abstract control theorem; it does not use the article's proposition as a premise.

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Sources

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