Elliptic and weakly coercive systems of operators in Sobolev spaces

D. V. Lymanskyi ; M. M. Malamud

Sbornik. Mathematics, Tome 199 (2008) no. 11, pp. 1649-1686

Résumé

It is known that an elliptic system $\{P_j(x,D)\}_1^N$ of order $l$ is weakly coercive in $\overset{\circ}{W}{}^l_{\!\infty}(\mathbb R^n)$, that is, all differential monomials of order $\leqslant l-1$ 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 $P(x,D)$ of $n\geqslant 3$ variables with constant principal part is weakly coercive in $\overset{\circ}{W}{}^l_{\!\infty}(\mathbb R^n)$ if and only if it is elliptic. A similar result is obtained for systems $\{P_j(D)\}_1^N$ with constant coefficients under the condition $n\geqslant 2N+1$ and with several restrictions on the symbols $P_j(\xi)$. A complete description of differential polynomials of two variables which are weakly coercive in $\overset{\circ}{W}{}^l_{\!\infty}(\mathbb R^2)$ is given. Wide classes of systems with constant coefficients which are weakly coercive in $\overset{\circ}{W}{}^l_{\!\infty}(\mathbb R^n)$, but non-elliptic are constructed. Bibliography: 32 titles.