Obstacle Problem

The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems. The problem is to find the equilibrium position of an elastic membrane whose boundary is held fixed, and which is constrained to lie above a given obstacle. It is deeply related to the study of minimal surfaces and the capacity of a set in potential theory as well. Applications include the study of fluid filtration in porous media, constrained heating, elasto-plasticity, optimal control, and financial mathematics.

The mathematical formulation of the problem is to seek minimizers of the Dirichlet energy functional,

in some domain where the functions represent the vertical displacement of the membrane. In addition to satisfying Dirichlet boundary conditions corresponding to the fixed boundary of the membrane, the functions are in addition constrained to be greater than some given obstacle function . The solution breaks down into a region where the solution is equal to the obstacle function, known as the contact set, and a region where the solution is above the obstacle. The interface between the two regions is the free boundary.

In general, the solution is continuous and possesses Lipschitz continuous first derivatives, but that the solution is generally discontinuous in the second derivatives across the free boundary. The free boundary is characterized as a Hölder continuous surface except at certain singular points, which reside on a smooth manifold.