Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 5 / 3 / 2 / a

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- Solution a

Solution

Write

R = Λ/ λ

. The coefficient is measurable, so interpret the divergence-form elliptic operator weakly:

\int A u^{'} ϕ^{'} = 0

for compactly supported

H^{1}

tests. Testing with

ϕ = ζ^{2} u

gives

λ \int ζ^{2} ∣ u^{'} ∣^{2} \leq 2Λ \int ζ ∣ u^{'} ∣∣ u ∣∣ ζ^{'} ∣ \leq 2Λ (\int ζ^{2} ∣ u^{'} ∣^{2})^{1/2} (\int u^{2} ∣ ζ^{'} ∣^{2})^{1/2} .

Divide if the first integral is positive; the zero case is immediate. Since

ζ = 1

on the inner interval,

\int_{- 1/2}^{1/2} ∣ u^{'} ∣^{2} \leq 4 R^{2} \int_{- 1}^{1} u^{2} ∣ ζ^{'} ∣^{2} .

This Caccioppoli inequality applies in particular to the smooth solutions in the question. No derivative of

A

is taken.

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