On Subordination Conditions for Systems of Minimal Differential Operators

In this paper, we provide a review of results on a priori estimates for systems of minimal differential operators in the scale of spaces L^p(Ω), where p ∈ [1, ∞]. We present results on the characterization of elliptic and l-quasielliptic systems using a priori estimates in isotropic and anisotropic Sobolev spaces \({W}_{p,0}^{l}\left({\mathbb{R}}^{n}\right)\) $W_{p,0}^l\left({\mathbb{R}}^n\right)$ , p ∈ [1, ∞]. For a given set l = (l₁, . . . , l_n) ∈ \({\mathbb{N}}^{n}\) ${\mathbb{N}}^n$ we prove criteria for the existence of l-quasielliptic and weakly coercive systems and indicate wide classes of weakly coercive in \({W}_{p,0}^{l}\left({\mathbb{R}}^{n}\right)\) $W_{p,0}^l\left({\mathbb{R}}^n\right)$ , p ∈ [1, ∞], nonelliptic, and nonquasielliptic systems. In addition, we describe linear spaces of operators that are subordinate in the L^∞( \({\mathbb{R}}^{n}\) ${\mathbb{R}}^n$ )-norm to the tensor product of two elliptic differential polynomials.