Poincaré Inequality Lambert M Surhone

Publisher Marketing: Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. High Quality Content by WIKIPEDIA articles!In mathematics, the Poincar inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincar . 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 the Friedrichs' inequality. The optimal constant C in the Poincar inequality is sometimes known as the Poincar constant for the domain . Determining the Poincar constant is, in general, a very hard task that depends upon the value of p and the geometry of the domain . Certain special cases are tractable, however. For example, if is a bounded, convex, Lipschitz domain with diameter d, then the Poincar constant is at most d/2 for p = 1, d/ for p = 2 (Acosta & Dur n 2004; Payne & Weinberger 1960), and this is the best possible estimate on the Poincar constant in terms of the diameter alone. In one dimension, this is Wirtinger's inequality for functions.