Poincaré inequality
From Wikipedia, the free encyclopedia
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.
[edit] Statement of the inequality
Assume that 
, Ω is precompact open subset of 
with Lipschitz boundary (i.e., Ω is an open, bounded Lipschitz domain). Then there exists a constant C, depending only on Ω and p, such that, for every 
,
where
is the average value of u over Ω.
[edit] Reference
- Evans, Lawrence C. (1998). Partial differential equations. Providence, RI: American Mathematical Society. ISBN 0-8218-0772-2.
