 Solution

ID: past-exam-of-the-fixed-point-methods-for-nonlinear-partial-differential-equations-course-of-the-university-of-oxford/2023/3/b/iii/solution

For

u, w \in X

⟨ A (u) - A (w), u - w ⟩ = \int_{Ω} (∣\nabla u ∣^{2} \nabla u - ∣\nabla w ∣^{2} \nabla w) \cdot (\nabla u - \nabla w) d x + \int_{Ω} (u^{7} - w^{7}) (u - w) d x .

s \mapsto s^{7}

is strictly increasing on all of

R

d = ξ - η

and

m = (ξ + η) /2

for real vectors

ξ, η

. Expanding gives

(∣ ξ ∣^{2} ξ - ∣ η ∣^{2} η) \cdot (ξ - η) = ∣ m ∣^{2} ∣ d ∣^{2} + 2 (m \cdot d)^{2} + \frac{1}{4} ∣ d ∣^{4} \geq \frac{1}{4} ∣ ξ - η ∣^{4} .

Consequently

⟨ A (u) - A (w), u - w ⟩ \geq \frac{1}{4} ∥\nabla (u - w) ∥_{4}^{4} = \frac{1}{4} ∥ u - w ∥_{X}^{4} > 0 (u \neq = w) .

u - w \in W_{0}^{1, 4} (Ω)

vanishes, then

u = w

. This proves that

A

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