Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 105 / 4 / a

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- Solution a

Solution

Extend

u

by zero to

R^{n}

; its compact support inside the ball makes the extension smooth. The Fourier transform of a derivative gives

L_{0} u (ξ) = - (A ξ \cdot ξ) u (ξ) .

The assumption says that

L_{0}

is a uniformly elliptic operator, so

∣ A ξ \cdot ξ ∣^{2} \geq θ^{2} ∣ ξ ∣^{4} .

Moreover,

\sum_{i, j = 1}^{n} ∣ ξ_{i} ξ_{j} ∣^{2} = (\sum_{i = 1}^{n} ξ_{i}^{2})^{2} = ∣ ξ ∣^{4} .

The Plancherel theorem therefore yields

∥ L_{0} u ∥_{2}^{2} = \int_{R^{n}} ∣ A ξ \cdot ξ ∣^{2} ∣ u ∣^{2} d ξ \geq θ^{2} \int_{R^{n}} ∣ ξ ∣^{4} ∣ u ∣^{2} d ξ = θ^{2} ∥ D^{2} u ∥_{2}^{2} .

All integrands vanish outside the original support where appropriate, so

θ ∥ D^{2} u ∥_{L^{2} (B_{r} (x_{0}))} \leq ∥ L_{0} u ∥_{L^{2} (B_{r} (x_{0}))} .

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