Formal Statement
Suppose the following data is given:
- an open bounded domain ⊂ ℝn with smooth boundary
- a smooth function on ∂ (the boundary of )
- 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.
Read more about this topic: Obstacle Problem
Famous quotes containing the words formal and/or statement:
“The bed is now as public as the dinner table and governed by the same rules of formal confrontation.”
—Angela Carter (1940–1992)
“Truth is that concordance of an abstract statement with the ideal limit towards which endless investigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth.”
—Charles Sanders Peirce (1839–1914)