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 manifestation of poetry in external life is formal perfection. True sentiment grows within, and art must represent internal phenomena externally.”
—Franz Grillparzer (1791–1872)
“If we do take statements to be the primary bearers of truth, there seems to be a very simple answer to the question, what is it for them to be true: for a statement to be true is for things to be as they are stated to be.”
—J.L. (John Langshaw)