Weighted Korn and Poincar\'e-Korn inequalities in the Euclidean space and associated operators

TL;DR

This paper establishes weighted Korn and Poincaré-Korn inequalities in Euclidean space with bounded measures satisfying Poincaré inequalities, analyzing associated operators and providing quantitative, constructive estimates relevant to kinetic theory and mechanics.

Contribution

It introduces new weighted Korn and Poincaré-Korn inequalities in Euclidean space with explicit constants and studies related self-adjoint operators, extending classical inequalities to weighted settings.

Findings

Derived weighted Korn inequalities with explicit constants.

Established Poincaré-Korn inequalities for vector field projections.

Connected inequalities with kinetic theory and classical mechanics.

Abstract

We prove functional inequalities on vector fields on the Euclidean space when it is equipped with a bounded measure that satisfies a Poincar\'e inequality, and study associated self-adjoint operators. The weighted Korn inequality compares the differential matrix, once projected orthogonally to certain finite-dimensional spaces, with its symmetric part and, in an improved form of the inequality, an additional term. We also consider Poincar\'e-Korn inequalities for estimating a projection of the vector field by the symmetric part of the differential matrix and zeroth-order versions of these inequalities obtained using the Witten-Laplace operator. The constants depend on geometric properties of the potential and the estimates are quantitative and constructive. These inequalities are motivated by kinetic theory and related with the Korn inequality (1906) in mechanics, on a bounded domain.