Elliptic and weakly coercive systems of operators in Sobolev spaces

TL;DR

This paper investigates the properties and conditions of weakly coercive elliptic systems of operators in Sobolev spaces, extending classical results and characterizing such systems with variable and constant coefficients.

Contribution

It establishes new criteria for weak coercivity, generalizes the de Leeuw-Mirkil theorem for variable coefficients, and classifies weakly coercive polynomials in two variables.

Findings

An analogue of the de Leeuw-Mirkil theorem for variable coefficient operators in n≥3.

Complete classification of weakly coercive polynomials in two variables.

Construction of wide classes of non-elliptic weakly coercive systems.

Abstract

It is known that an elliptic system $\{P_j(x,D)\}_1^N$ of order $l$ is weakly coercive in $\overset{\circ}{W}\rule{0pt}{2mm}^l_\infty(\mathbb R^n)$, that is, all differential monomials of order $\le l-1$ on $C_0^\infty(\mathbb R^n)$-functions are subordinated to this system in the $L^\infty$-norm. Conditions for the converse result are found and other properties of weakly coercive systems are investigated. An analogue of the de Leeuw-Mirkil theorem is obtained for operators with variable coefficients: it is shown that an operator $P(x,D)$ in $n\ge 3$ variables with constant principal part is weakly coercive in $\overset{\circ}{W}\rule{0pt}{2mm}_\infty^l(\mathbb R^n)$ if and only if it is elliptic. A similar result is obtained for systems $\{P_j(x,D)\}_1^N$ with constant coefficients under the condition $n\ge 2N+1$ and with several restrictions on the symbols $P_j(\xi)$ . A complete description of differential polynomials in two variables which are weakly coercive in $\overset{\circ}{W}\rule{0pt}{2mm}_\infty^l(\mathbb R^2)$ is given. Wide classes of systems with constant coefficients which are weakly coercive in $\overset{\circ}{W}\rule{0pt}{2mm}_\infty^l(\mathbb \R^n)$, but non-elliptic are constructed.