The functional dissipativity of certain systems of partial differential equations

TL;DR

This paper investigates the conditions under which certain systems of partial differential equations exhibit functional dissipativity, providing algebraic criteria and exploring the relationship with functional ellipticity.

Contribution

It establishes algebraic necessary and sufficient conditions for the functional dissipativity of specific PDE systems and examines different notions of functional ellipticity.

Findings

Derived algebraic criteria for dissipativity

Analyzed relations between dissipativity and ellipticity

Provided conditions for systems with complex matrix coefficients

Abstract

In the present paper we consider the functional dissipativity of the Dirichlet problem for systems of partial differential operators of the form $\partial_{h} ({\mathop{\mathscr A}\nolimits}^{hk}(x)\partial_{k})$ (${\mathop{\mathscr A}\nolimits}^{hk}$ being $m\times m$ matrices with complex valued $L^{1}_{\text{loc}}$ entries). In the particular case of the operator $\partial_{h} ({\mathop{\mathscr A}\nolimits}^{h}(x)\partial_{h})$ (where ${\mathop{\mathscr A}\nolimits}^{h}$ are $m\times m$ matrices) we obtain algebraic necessary and sufficient conditions. We give also three different notions of functional ellipticity and investigate the relations between them and the functional dissipativity for the operators in question.