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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-9/6/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 9 / 6 / b / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

Fix

x

with

dist (x, \partial Ω) > σ

and use

η_{x} (y) = ϕ_{σ} (x - y)

as the test function in part (a). Its support lies compactly inside

Ω

. Differentiating the convolution under the integral is allowed, and

D_{y_{i}} η_{x} = - D_{x_{i}} ϕ_{σ} (x - y)

while

Δ_{y} η_{x} = Δ_{x} ϕ_{σ} (x - y)

. Consequently

Δ u_{σ} (x) = \int_{Ω} u (y) Δ_{y} η_{x} (y) d y = - i \sum \int_{Ω} b^{i} (y) u (y) D_{y_{i}} η_{x} (y) d y + \int_{Ω} (c u + f) (y) η_{x} (y) d y = i \sum D_{x_{i}} \int_{Ω} b^{i} (y) u (y) ϕ_{σ} (x - y) d y + (c u)_{σ} (x) + f_{σ} (x) .

Thus

Δ u_{σ} = i \sum D_{i} (b^{i} u)_{σ} + (c u)_{σ} + f_{σ}

at every stated interior point. This is the convolution derivative identity. The convolutions are local, so no integrability of the smooth coefficients all the way to the boundary is needed. In particular,

(b^{i} u)_{σ}

is the convolution of the product; it must not be replaced by

b^{i} u_{σ}

for a variable coefficient.

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