Elliptic and weakly coercive systems of operators in Sobolev spaces

Abstract

It is known that an elliptic system $\{P_j(x,D){\}}_1^N$ of order $l$ is weakly coercive in $\stackrel{\circ }{W}{}_{\infty }^l({\mathbb{R}}^n)$, 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 $\stackrel{\circ }{W}{}_{\infty }^l({\mathbb{R}}^n)$ 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 $\stackrel{\circ }{W}{}_{\infty }^l({\mathbb{R}}^2)$ is given. Wide classes of systems with constant coefficients which are weakly coercive in $\stackrel{\circ }{W}{}_{\infty }^l({\mathbb{R}}^n)$, but non-elliptic are constructed.

36 pages, 1 figure