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

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