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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-4/23h/b/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 23H / b / i / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

Weak convergence

u_{n} ⇀ u

in

H_{0}^{1} (Ω)

implies

D u_{n} ⇀ D u

in

L^{2}

. The

L^{2}

norm is weakly lower semicontinuous, so

\int_{Ω} ∣ D u ∣^{2} \leq lim inf_{n} \int_{Ω} ∣ D u_{n} ∣^{2} .

The Rellich-Kondrashov compactness theorem for H01 upgrades the weak convergence to

u_{n} ⟶ u

strongly in

L^{2} (Ω)

. Indeed, compactness gives this along every subsequence after passage to a further subsequence, and the weak limit uniquely identifies every such strong limit as

u

. Consequently,

\int_{Ω} V (u_{n}^{2} - u^{2}) \leq ∥ V ∥_{\infty} ∥ u_{n} - u ∥_{2} (∥ u_{n} ∥_{2} + ∥ u ∥_{2}) ⟶ 0.

Combining the two terms proves the weak lower semicontinuity of a bounded-domain Schrodinger energy:

E (u) \leq n lim inf E (u_{n}) .

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