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 spiritual kinship between Lincoln and Whitman was founded upon their Americanism, their essential Westernism. Whitman had grown up without much formal education; Lincoln had scarcely any education. One had become the notable poet of the day; one the orator of the Gettsyburg Address. It was inevitable that Whitman as a poet should turn with a feeling of kinship to Lincoln, and even without any association or contact feel that Lincoln was his.”
—Edgar Lee Masters (1869–1950)
“After the first powerful plain manifesto
The black statement of pistons, without more fuss
But gliding like a queen, she leaves the station.”
—Stephen Spender (1909–1995)