Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 107 / 4 / a / i

Table of contents
- Solution i

Solution

Multiply

Δ u + q u \leq 0

by

ζ^{2} / u_{ε}

and integrate. Since

ζ

is compactly supported, integration by parts and Young's inequality give

\int q \frac{u}{u _{ε}} ζ^{2} \leq - \int \frac{Δ u}{u _{ε}} ζ^{2} = 2 \int \frac{ζ D u \cdot D ζ}{u _{ε}} - \int \frac{ζ ^{2} ∣ D u ∣ ^{2}}{u _{ε}^{2}} \leq \int ∣ D ζ ∣^{2} .

ε ↓ 0

,

u / u_{ε}

converges to the indicator of

{u > 0}

. The dominated convergence theorem therefore gives

\int_{{u > 0}} q ζ^{2} \leq \int ∣ D ζ ∣^{2} .

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