Obstacle problem

Remarks

The set $K$ is nonempty, closed and convex in $H^1_0(\Omega)$, since it contains $\max(\psi, 0)$. So the problem has exactly one solution, by the Stampacchia theorem for variational inequalities and the Poincaré inequality. Equivalently, $u \in K$ satisfies $\int_\Omega \nabla u \cdot \nabla (v - u) \, dx \ge 0$ for all $v \in K$. When $u$ is regular enough, this is the complementarity system $u \ge \psi$, $\Delta u \le 0$, and $\Delta u = 0$ where $u > \psi$. So $u$ is harmonic in the open set $\set{u > \psi}$ and equals $\psi$ on the contact set $\set{u = \psi}$. The boundary of the contact set inside $\Omega$ is not known in advance, so this is a free boundary problem. If $\psi$ is smooth, the solution has bounded second derivatives locally, and this is the best regularity possible in general, since $\Delta u$ jumps across the free boundary. In dimension one with $\Omega = (-1, 1)$, the solution is the smallest concave function above $\psi$ that vanishes at $\pm 1$: it is linear off the contact set.