Weighted Korn and Poincaré-Korn Inequalities in the Euclidean Space and Associated Operators

We prove functional inequalities on vector fields u: R^d→ R^d when R^d is equipped with a bounded measure e-ϕdx that satisfies a Poincaré inequality, and study associated self-adjoint operators. The weighted Korn inequality compares the differential matrix Du, once projected orthogonally to certain finite-dimensional spaces, with its symmetric part D^su and, in an improved form of the inequality, an additional term ∇ ϕ· u. We also consider Poincaré-Korn inequalities for estimating a projection of u by D^su and zeroth-order versions of these inequalities obtained using the Witten-Laplace operator. The constants depend on geometric properties of the potential ϕ and the estimates are quantitative and constructive. These inequalities are motivated by kinetic theory and related with the Korn inequality (1906) in mechanics, which compares Du and D^su on a bounded domain.