Elliptic complexes and generalized Poincar\'e inequalities

TL;DR

This paper investigates conditions under which generalized Poincaré inequalities hold for first order differential operators with constant coefficients, establishing the sufficiency of the constant rank condition using Moore-Penrose inverses.

Contribution

It proves that the constant rank condition ensures generalized Poincaré inequalities for certain differential operators, linking matrix theory with PDE analysis.

Findings

Constant rank condition is sufficient for generalized Poincaré inequalities.

Utilizes Moore-Penrose inverse to analyze differential operators.

Provides a criterion connecting matrix properties to PDE inequalities.

Abstract

We study first order differential operators with constant coefficients. The main question is under what conditions a generalized Poincar\'e inequality holds. We show that the constant rank condition is sufficient. The concept of the Moore-Penrose generalized inverse of a matrix comes into play.