A new approach to the theory of linear problems for systems of differential partial equations. II
Part I [V. S. Mokeĭchev and A. V. Mokeĭchev, Izv. Vyssh. Uchebn. Zaved. Mat. 1999, no. 1, 25--35; Zbl 1049.35071)] contained Section 1 and part of Section 2 of the complete work. The second part starts from Theorem 2.3. If the condition (A) of Part I holds, then the following propositions are equivalent: (1) If for all \(f(x)\) there exists a \(\varphi\)-solution \(u\) of \(\bigl (P(x,\,\partial_ {x})-\lambda E \bigr )u=f(x)\) in \(H,\,\,x\in \Omega \subset \mathbb{R}^ {n}\) and \(lu=0\), then \(u\) is differentiable up to order \(\beta\) in \(H(\beta)\). (2) For all \(\psi \in L_ {\varphi}\), there exists \(c_{\psi}>0\), \(N\) and \(c(N)\) independent of \(\psi\) such that \(c_{\psi }\|\psi ^{(\beta)}\|_{\beta} -c(N)\|\psi (N,x)\|\leq \|Q \psi \|\). This is a global regularity theorem; another local regularity theorem is also given. Section 3 concerns the relation between a \(\varphi\)-solution and a \(\tilde {\varphi}\)-solution of the same problem. Theorem 3.1: Let the domains of the operators \(P_{\lambda}\), \(Q\), \(\tilde {P_{\lambda}}\) and \(\tilde {Q}\) satisfy the relations \({\mathcal D} _{P_{\lambda}}={\mathcal D}_ {Q}\), \({\mathcal D}_{\tilde {P_{\lambda}}}= {\mathcal D}_{\tilde{Q_{\lambda}}}\), and let \(\varphi _{(1)}\) and \(\tilde {\varphi _{(k)}}\) be orthogonal systems of \(H_{(0)}\) such that for all \(k \in N\), \(P_{\lambda} \varphi _{(p)}=\sum a_{p,k} \tilde{P_{\lambda}} \tilde {\varphi _{(k)}}\). Then a \(\varphi\)-solution \(u\) can be extended to a \(\tilde {\varphi}\)-solution of the same problem, and both are identical if \({\mathcal D}_ {Q} \subset H_{0}, {\mathcal D} _{\tilde Q}\subset H_{0} \). Section 4 concerns the correctly solvable problems for differential quasi-polynomials with constant coefficients. In Section 5 the authors consider the equation \[ \begin{multlined} P_{1}(x,\,\partial _{x})=\\\sum _ {\alpha \in \Phi _{(2)}} \Big [a _{\alpha}(x)\bigl ((\partial _{x})^ {2}+h \bigr )^ {\alpha}+ \sum _ {\beta <\alpha} b _{\alpha ,\beta} (x)(\partial _{x})^{\gamma (\alpha ,\beta)} \bigl ((\partial _{x})^{2} +h \bigr )^ {\beta} \Big]\\-\lambda u=f(x),\quad x \in \mathbb R^{n}.\end{multlined} \] They also provide a rather technical theorem (Theorem 5.1) on the condition for the operator \(P\) possessing only a point-spectrum, and some estimates of the Fourier coefficients of the \(\varphi\)-solution. For Part III, cf. Russ. Math. 43, No. 11, 47--57 (1999); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1999, No. 11, 50--59 (1999; Zbl 1049.35073).
scientific article; zbMATH DE number 1541500