Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 107 / 3 / b / ii

Table of contents
- Solution ii

Solution

Differentiate the equation for

v_{R}

with respect to

x_{k}

. The derivative

w = D_{k} v_{R}

is a weak solution of

D_{i} (a^{ij} (D v (R x)) D_{j} w) = 0,

where

a^{ij} (p) = \frac{δ ^{ij}}{1 + ∣ p ∣ ^{2}} - \frac{p _{i} p _{j}}{( 1 + ∣ p ∣ ^{2} ) ^{3/2}} .

The eigenvalue in the direction of

p

is

(1 + ∣ p ∣^{2})^{- 3/2}

and every orthogonal eigenvalue is

(1 + ∣ p ∣^{2})^{- 1/2}

. Thus a uniform bound on

∣ D v ∣

makes this a uniformly elliptic operator with constants independent of

R

.

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