Summary: OPTIMAL COERCIVITY INEQUALITIES IN W 1,p
Abstract. This paper describes the characterization of optimal constants for some
coercivity inequalities in W 1,p
(), 1 < p < . A general result involving inequalities
of p-homogeneous forms on a reflexive Banach space is first proved. The constants
are shown to be the least eigenvalues of certain eigenproblems with equality holding
for the corresponding eigenfunctions. This result is applied to 3 different classes of
coercivity results on W 1,p
(). The inequalities include very general versions of the
Friedrichs' and PoincarŽe inequalities. Scaling laws for the inequalities are also given.
This paper describes the characterization of optimal constants, and corresponding
optimal functions, for some inequalities satisfied by functions in the Sobolev spaces
(), 1 < p < . Here is a bounded, connected, open set in Rn
assumptions (A1) and (A2) of section 2.
First some general properties of p-homogeneous inequalities on a reflexive Banach