Green's formula and theorems on complete collection of isomorphisms for general elliptic boundary value problems for Douglis-Nirenberg systems (Q2735864)

In the bounded domain \(G\subset \mathbb{R}^n\) with boundary \(\partial G\in C^\infty\) the authors consider the elliptic boundary value problem for Douglis-Nirenberg elliptic system NEWLINE\[NEWLINE\begin{aligned} l(x,D)u(x) &= (l_{rj} (x,D))_{r,j=1,\dots,N} u(x)= f(x), \quad x\in G, \tag{1}\\ b(x,D) u(x) &= (b_{hj}(x,D)) |_{{h=1,\dots, m}\atop {j=1,\dots, N}} u(x)= \varphi(x), \quad x\in \partial G. \tag{2} \end{aligned}NEWLINE\]NEWLINE For such general problems Green's formula is obtained, in which only scalar product in \((L_2(G))^N\) and \(L_2(\partial G)\) is used. With some additional assumptions this formula does not contain the projection operators. Using this formula the boundary problem formally adjoint to (1), (2) is introduced, which also turns out to be elliptic.NEWLINENEWLINEIn the corresponding functional spaces problem (1), (2) generates the Noetherian operator \(A\) and its solvability conditions are formulated with help the above mentioned Green's formula. Proofs of several theorems about isomorphisms which are realized by the operator \(A\) in various families of functional spaces are outlined. There are some misprints.