Mathematical inequality in Sobolev space theory
In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great importance in the modern, direct methods of the calculus of variations. A very closely related result is Friedrichs' inequality.
Statement of the inequality.
The classical Poincaré inequality.
Let "p", so that 1 ≤ "p"  ; ∞ and Ω a subset bounded at least in one direction. Then there exists a constant "C", depending only on Ω and "p", so that, for every function "u" of the Sobolev space "W"01,"p"(Ω) of zero-trace (a.k.a. zero on the boundary) functions,
formula_1
Poincaré–Wirtinger inequality.
Assume that 1 ≤ "p" ≤ ∞ and that Ω is a bounded connected open subset of the "n"-dimensional Euclidean space formula_2 with a Lipschitz boundary (i.e., Ω is a Lipschitz domain). Then there exists a constant "C", depending only on Ω and "p", such that for every function "u" in the Sobolev space "W"1,"p"(Ω),
formula_3
where
formula_4
It follows that the constant formula_5 is given as formula_6, however, the reference with Theorem 1 indicates that this is not the optimal constant.
Poincaré on Balls.
We can derive a growth constant for Balls in a manner similar to previous cases. The relationship is given by the following inequality:
formula_7
Sketch of the Proof.
The proof proceeds similarly to the classical one, by using the scaling formula_8. Then, by using a form of chain rule for the fractional derivative, we get formula_9 as a result.
References.
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