= Log-cosh obstacle variational inequality
{title2=$\int\nabla u\cdot\nabla(v-u)+\tanh(u)(v-u)\ge0$}
Minimizing the log-cosh energy over <zero-boundary Sobolev space>[zero-trace] <functions> above an obstacle yields the displayed inequality for every feasible competitor. <Convexity> supplies the one-sided admissible segments. On the open noncontact region, compactly supported variations of both signs are feasible because the gap from the obstacle has a positive minimum on their support. Their <first variations> vanish, giving $-\Delta u+\tanh u=0$ there. This stationarity equation need not hold on the contact <set>.
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