On iterations of the Green integrals and their applications to elliptic differential complexes.

*(English)*Zbl 0871.35066Let \(P\) be a linear partial differential operator of order \(p\geq 1\). The equation (1) \(Pu=f\) is considered for the operator \(P\) with injective symbol. The study is based on the concept of the Green integrals and Green operators, which are described. In particular, there is given a representation of functions from \(W^{p,2}\) through the Green operator and a left fundamental solution of \(P\). The theorem on iterations of the Green integrals is established. This allows to get the solvability conditions for the equation (1). Further, the first Sobolev cohomology group of elliptic differential complexes is studied, in particular, some criteria for its vanishing are obtained. Applications to the Neumann problem as well as to the Cauchy and Dirichlet problems related to the operator \(P\) and some concrete examples are given.

Reviewer: M.Shapiro (Mexico)

##### MSC:

35N10 | Overdetermined systems of PDEs with variable coefficients |

35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |

35N15 | \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs |

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\textit{M. Nacinovich} and \textit{A. Shlapunov}, Math. Nachr. 180, 243--284 (1996; Zbl 0871.35066)

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