 Obstacle problem

A classical obstacle problem minimizes the Dirichlet energy with forcing while requiring the unknown function to stay above an obstacle. On a bounded Lipschitz domain $\Omega$, take $f\in L^2(\Omega )$ and an obstacle $\psi \in H_0^1(\Omega )$. MinimizeThe closed convex set $K$ is nonempty, and the Poincaré inequality makes $J$ coercive and strictly convex. There is a unique minimizer $u$, characterized by the variational inequalityIt follows by differentiating $J(u+t(v-u))$ at $t=0^{+}$; convexity also makes the inequality sufficient for minimality.

Where regularity permits pointwise notation, the same conditions read $u\ge \psi$, $-\Delta u-f\ge 0$, and $(u-\psi )(-\Delta u-f)=0$. On the noncontact set, variations of both signs recover the unconstrained Poisson equation. On the contact set of an obstacle problem, the obstacle can supply a reaction force. Its interface is an unknown free boundary problem. An analogous separation into stopping and continuation regions appears in optimal stopping, where the value must dominate the immediate payoff.

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.