## 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)