 
Summary: OPTIMAL COERCIVITY INEQUALITIES IN W 1,p
().
GILES AUCHMUTY
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 phomogeneous 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.
1. Introduction
This paper describes the characterization of optimal constants, and corresponding
optimal functions, for some inequalities satisfied by functions in the Sobolev spaces
W1,p
(), 1 < p < . Here is a bounded, connected, open set in Rn
which satisfy
assumptions (A1) and (A2) of section 2.
First some general properties of phomogeneous inequalities on a reflexive Banach
