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A remark on ellipticity of systems of linear partial differential equations with constant coefficients. (English) Zbl 0532.35008
Astérisque 89-90 117-128 (1981).
The author considers matrix partial differential operators \(P(D)=(P_{jk}(D))\) (1\(\leq j\leq J;1\leq k\leq K)\) with constant coefficients. They are assumed to act from \(L^ p_{r,{\bar \Omega}}\) to \(L^ p_{s,{\bar \Omega}}\) (r,s, multi-indices, \(r_ k-s_ j\geq 0)\). Here \(L^ p_{t,{\bar \Omega}}=\prod^{m}_{j=1}L^ p_{t_ j,{\bar \Omega}}\), \(L^ p_{t_ j,{\bar \Omega}}\) denoting the usual space of Bessel potentials of \(L^ p\)-functions related to a bounded convex open set \(\Omega \subset R^ n\). Additional conditions to be imposed upon the (matrix) polynomial P are looked for in order that the following proposition holds: (1) \(P(D):L^ p_{r,{\bar \Omega}}\to L^ p_{s,{\bar \Omega}}\) is bounded with closed range. The author shows that (1) is equivalent to the ellipticity of P(D) in the sense of Douglis-Nirenberg, when P(D) is restricted to the class of determined operators (i.e. \(P(D)u=0\), \(u\in({\mathcal E}')^ K\Leftrightarrow u=0)\). For non-determined operators he proves that (1) holds when the ellipticity condition is replaced by the following ”very strong” ellipticity condition for \({}^ tP(D)=^ t(P_{jk}(-D)):\) rank\((\overset \circ P_{kj}(-\zeta))=J,\quad \forall \zeta \in C^ n\backslash \{0\},\quad \overset \circ P_{jk}\) denoting the part in \(P_{jk}\) of degree \(r_ k-s_ j\). Finally the author shows that, in the case where P(D) is of tensor product type (i.e. \(P(D)=(P_ 1(D'),...,P_ I(D')\), \(P_{I+1}(D''),...,P_ J(D''))\), \(D=(D',D''))\) and \(\Omega\) is of the form \(\Omega =\Omega '\times \Omega ''\) with \(\Omega\) ’ and \(\Omega\) ” bounded and convex open sets, then \({}^ tP(D)\) is very strongly elliptic.
For the entire collection see [Zbl 0481.00013].
Reviewer: A.Lorenzi
MSC:
35E20 General theory of PDEs and systems of PDEs with constant coefficients
35G05 Linear higher-order PDEs
35J45 Systems of elliptic equations, general (MSC2000)