Elliptic and weakly coercive systems of operators in Sobolev spaces

Details Elliptic and weakly coercive systems of operators in Sobolev spaces Elliptic and weakly coercive systems of operators in Sobolev spaces

2009-04-19

arxiv.org/abs/0904.2922v1

D.V. Limanskii, M.M. Malamud, Elliptic and weakly coercive systems of operators in Sobolev spaces Sbornik: Mathematics, 199: 1

Mathematics

Analysis of PDEs

36 pages, 1 figure

Scientific paper

10.1070/SM2008v199n11BEH003976

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.

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