Obstacle Problem - Formal Statement

Formal Statement

Suppose the following data is given:

1. an open bounded domain ⊂ ℝn with smooth boundary
2. a smooth function on ∂ (the boundary of )
3. a smooth function defined on all of such that <, i.e. the restriction of to the boundary of (its trace) is less than .

Then consider the set

which is a closed convex subset of the Sobolev space of square integrable functions with square integrable weak first derivatives, containing precisely those functions with the desired boundary conditions which are also above the obstacle. The solution to the obstacle problem is the function which minimizes the energy integral

over all functions belonging to ; the existence of such a minimizer is assured by considerations of Hilbert space theory.