Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 107 / 3

Table of contents
- a 3
  - Solution a
- b 3
  - Solution b

a

Solution

The Campanato space

L^{p, μ} (Ω)

consists of

f \in L^{p} (Ω)

for which

[f]_{L^{p, μ}}^{p} = sup_{x_{0}, r} r^{- μ} \int_{Ω \cap B_{r} (x_{0})} ∣ f - f_{Ω \cap B_{r} (x_{0})} ∣^{p} < \infty,

where

f_{E} = ∣ E ∣^{- 1} \int_{E} f

. On a smooth bounded domain,

L^{p, n + p α} (Ω) = C^{0, α} (\overline{Ω})

with equivalent norms for

0 < α \leq 1

.

Solved by gpt-5.6-sol high.

b

Solution

Fix

B_{R} (x_{0}) \subset B_{1}

, put

A_{0} = A (x_{0})

, and let

v

have the same boundary values as

u

while solving

div (A_{0} \nabla v) = 0

. Then

w = u - v \in W_{0}^{1, 2} (B_{R})

satisfies

div (A_{0} \nabla w) = div ((A_{0} - A (x)) \nabla u) .

Since

u \in C^{1}

gives

∣ D u ∣ \leq L

locally, the supplied energy estimate and Hölder continuity of

A

yield

\int_{B_{R}} ∣\nabla w ∣^{2} \leq C L^{2} [A]_{C^{α}}^{2} R^{n + 2 α} .

The constant-coefficient decay estimate gives

\int_{B_{ρ}} ∣\nabla v - (\nabla v)_{B_{ρ}} ∣^{2} \leq C (ρ / R)^{n + 2} \int_{B_{R}} ∣\nabla v - (\nabla v)_{B_{R}} ∣^{2} .

Thus, for

Φ (r) = \int_{B_{r}} ∣\nabla u - (\nabla u)_{B_{r}} ∣^{2}

,

Φ (ρ) \leq C (ρ / R)^{n + 2} Φ (R) + C R^{n + 2 α} .

Because

n + 2 > n + 2 α

, the Campanato iteration lemma gives

Φ (ρ) \leq C ρ^{n + 2 α}

on balls in

B_{1/2}

. Hence

\nabla u \in L^{2, n + 2 α} = C^{0, α}

and

u \in C^{1, α} (B_{1/2})

.

Solved by gpt-5.6-sol high.

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