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Strong $L^{2}$ convergence preserves an $L^{2}$ -normalisation constraint

Example

Assume the Axiom of Choice. Let $n \geq 1$ , let $Ω \subseteq R^{n}$ be a bounded extension domain and let $u_{j} ⇀ u$ weakly in $H^{1} (Ω)$ with $\sup_{j} ∥ u_{j} ∥_{H^{1} (Ω)} < \infty$ and $∥ u_{j} ∥_{L^{2} (Ω)} = 1$ for all $j$ . Then $u_{j} \to u$ in $L^{2} (Ω)$ and $∥ u ∥_{L^{2} (Ω)} = 1$ . Thus an $L^{2}$ -normalisation constraint passes to the weak limit, which is exactly the step used when a constrained minimisation or eigenvalue problem is solved by taking a weakly convergent minimising sequence and then upgrading to strong convergence.

Facts & Assumptions

Given: the Axiom of Choice, a bounded extension domain $Ω \subseteq R^{n}$ , a sequence $u_{j} ⇀ u$ weakly in $H^{1} (Ω)$ with $\sup_{j} ∥ u_{j} ∥_{H^{1} (Ω)} < \infty$ and $∥ u_{j} ∥_{L^{2} (Ω)} = 1$ .

[F1]

Strong convergence upgrades the weak limit. $u_{j} \to u$ in $L^{2} (Ω)$ . (Weak $H^{1}$ convergence plus compactness gives strong $L^{2}$ convergence, Weak convergence of nets and sequences, The notation $H^{k}$ and the reserved zero-boundary symbol)

[F2]

The reverse triangle inequality. $∣ ∥ g ∥ - ∥ h ∥ ∣ \leq ∥ g - h ∥$ for every norm, in particular for the $L^{2}$ norm. Indeed $∥ g ∥ \leq ∥ g - h ∥ + ∥ h ∥$ and the exchanged inequality follow from the norm triangle inequality. (The space $L^{p} (μ)$ as the quotient by null functions)

Verification

technique · direct

1.1F1F2given

By [F1] the sequence converges strongly in $L^{2} (Ω)$ ; by [F2] applied to the $L^{2}$ norm, $∣ ∥ u_{j} ∥_{L^{2}} - ∥ u ∥_{L^{2}} ∣ \leq ∥ u_{j} - u ∥_{L^{2}} \to 0$ .

2.1F1F2step 1.1∎

Since $∥ u_{j} ∥_{L^{2}} = 1$ for every $j$ , step 1.1 forces $∥ u ∥_{L^{2}} = 1$ ; hence the weak limit of a normalised sequence is again normalised and lies in the constraint set ${v : ∥ v ∥_{L^{2}} = 1}$ , so it is an admissible candidate for a constrained minimiser. No weak lower semicontinuity of any energy is asserted here; only the passage of the normalisation to the limit is. The Axiom of Choice is inherited through [F1].

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Sources

Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, complete graduate notes) (standard reference, not scraped)

Strong L2L2 convergence preserves an L2L2-normalisation constraint