 Solution

ID: past-exam-of-the-fixed-point-methods-for-nonlinear-partial-differential-equations-course-of-the-university-of-oxford/2008/3/b/solution

X = H_{0}^{1} (Ω)

with norm

∥ u ∥_{X} = ∥\nabla u ∥_{2}

. The Poincaré inequality makes this equivalent to the usual Sobolev norm. Define

⟨ A (u), v ⟩ = \int_{Ω} \nabla u \cdot \nabla v + \int_{Ω} u^{3} v, ⟨ f, v ⟩ = \int_{Ω} v .

u \in X

and

⟨ A (u), v ⟩ = ⟨ f, v ⟩

for every

v \in X

∥ u ∥_{6} \leq C_{6} ∥ u ∥_{X}

, so

u^{3} \in L^{2}

∣ ⟨ A (u), v ⟩ ∣ \leq (∥ u ∥_{X} + C_{P} C_{6}^{3} ∥ u ∥_{X}^{3}) ∥ v ∥_{X}, ∣ ⟨ f, v ⟩ ∣ \leq ∣Ω ∣^{1/2} C_{P} ∥ v ∥_{X} .

Thus

A (u) \in X^{*}

for every

u

, and

f \in X^{*}

. Finally

⟨ A (u) - A (w), u - w ⟩ = ∥ u - w ∥_{X}^{2} + \int_{Ω} (u - w)^{2} (u^{2} + u w + w^{2}) \geq ∥ u - w ∥_{X}^{2},

since

u^{2} + u w + w^{2} \geq 0

. Hence

A

is strongly monotone with constant one in the chosen norm.

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