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

Abstract

We prove functional inequalities on vector fields \(u: {{\mathbb {R}}}^d \rightarrow {{\mathbb {R}}}^d\) when \({{\mathbb {R}}}^d\) is equipped with a bounded measure \(\hbox {e}^{-\phi } \,\mathrm {d}x\) 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 \(D^s u\) and, in an improved form of the inequality, an additional term \(\nabla \phi \cdot u\). We also consider Poincaré-Korn inequalities for estimating a projection of u by \(D^s u\) and zeroth-order versions of these inequalities obtained using the Witten-Laplace operator. The constants depend on geometric properties of the potential \(\phi \) 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 \(D^s u\) on a bounded domain.