An obstacle constraint requires a solution to lie above a given function. In an optimal stopping problem, the value dominates the payoff, satisfies a generator equation where continuation is optimal, and a generator inequality where stopping is optimal. The complementary conditions can be expressed as a maximum equal to zero.
The noncontact set is where the unknown function is strictly above its obstacle. For continuous functions it is open. Every smooth variation with compact support in this set remains feasible with either sign when its amplitude is small enough. The first variation therefore vanishes there, yielding the unconstrained Euler-Lagrange equation. On the complementary contact set of an obstacle problem, only the appropriate one-sided variational inequality is available globally.
The contact set is the region where the obstacle inequality is active. On its complement the obstacle supplies no reaction; its boundary is a free boundary, determined along with the unknown solution. A point can belong to the contact set while its contact force is zero, as for a taut horizontal string just touching an obstacle at one point. Thus equality of the gap alone does not imply strictly positive pressure.
For a string of positive tension with fixed endpoints , zero transverse loading and obstacle , , equilibrium minimizes among functions above . On a free interval , while in contact . Smooth tangency of the right line through to the parabola at requires , yielding the displayed root in .
The two pieces join with equal value and slope. On the free portions ; in contact is the nonnegative reaction. These facts verify the complementarity conditions. The associated variational inequality and the exact energy difference prove global minimality; strict convexity with fixed endpoints proves uniqueness. At contact reduces to one point; at it fills the whole interval.
Minimizing the log-cosh energy over 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 there. This stationarity equation need not hold on the contact set.
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The "obstacle problem" typically refers to a type of variational problem in which one studies the properties of a function that satisfies certain conditions while being constrained by obstacles in its domain. More formally, it often pertains to finding the minimum of a functional subject to certain constraints represented by obstacles.