The statement of well-posed problems for degenerate systems of partial differential equations

Let ∝ⁿ as usual denote the n-dimensional Cartesian space of points x = (x₁,…,x_n). In some domain Ω φ ℞ⁿ we consider a system of linear partial differential equations with respect to the vector function u(x) = (u₁,…,u_m) 6.1https://www.w3.org/1998/Math/MathML">∑|α|≤NAα(x)Dαu=F(x),https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062233/9da0ee55-ab85-4900-bd1b-460e01b2ece9/content/eqn6_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where A_α(x) are (r × m) matrices and F(x) = (F₁,…,F_m) is a vector function given in Ω. We presuppose for (6.1) to be degenerate in Ω, i.e. even in case r = m, 6.2https://www.w3.org/1998/Math/MathML">detσ(x,ξ)≡0, σ(x,ξ)=∑|α|≤NAα(x)ξα,https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062233/9da0ee55-ab85-4900-bd1b-460e01b2ece9/content/eqn6_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for any x ∈ Ω and any ξ = (ξ₁,…,ξ_n) ∈ ℝⁿ.