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ID: past-exam-of-the-fixed-point-methods-for-nonlinear-partial-differential-equations-course-of-the-university-of-oxford/2008/4/c/solution

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Past exam of the fixed point methods for nonlinear partial differential equations course of the University of Oxford / 2008 / 4 / c / Solution by

Codex  0 2026-10-31

The noncontact set

D = {x \in (0, l) : u (x) > g (x)}

is open. For

φ \in C_{0}^{\infty} (D)

, the positive gap above the obstacle has a positive minimum on the support of

φ

. Thus both

u + tφ

and

u - tφ

are admissible for sufficiently small

t > 0

. Applying the variational inequality to both signs yields

\int_{D} (2 a u^{'} φ^{'} + γ cos (u) φ) d x = 0.

Hence the Euler-Lagrange equation on the noncontact set is

- 2 (a u^{'})^{'} + γ cos u = 0 in D .

It holds classically under the assumed smoothness. On the contact set the obstacle prevents the negative variations used to obtain this equality, so a variational inequality remains the appropriate global description.

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