= Solution
The <noncontact set> $D=\{x\in(0,l):u(x)>g(x)\}$ is open. For $\varphi\in C_0^\infty(D)$, the positive gap above the obstacle has a positive minimum on the support of $\varphi$. Thus both $u+t\varphi$ and $u-t\varphi$ are admissible for sufficiently small $t>0$. Applying the <variational inequality> to both signs yields
$$
\int_D\bigl(2au'\varphi'+\gamma\cos(u)\varphi\bigr)\,dx=0.
$$
Hence the <Euler-Lagrange equation> on the noncontact set is
$$
\boxed{-2(au')'+\gamma\cos u=0\quad\text{in }D.}
$$
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.
Back to article page
