# zbMATH — the first resource for mathematics

On iterations of the Green integrals and their applications to elliptic differential complexes. (English) Zbl 0871.35066
Let $$P$$ 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
Full Text:
##### References:
 [1] Aizenberg , L. A. Yuzhakov , A. P. 1979 [2] Andreotti, On Analytic and C Poncaré Lemma, Mathematical Analysis and Applications, Advances in Math. Suppl. Studies 7A pp 41– (1981) [3] Henkin, Method of Integral Representations in Complex Analysis, Results of Sciences and Technics, Modern Problems of Mathematics. Fundamental Trends, VINITI 7 pp 23– (1985) [4] Henkin, Theory of Functions on Complex Manifolds (1984) [5] Hörmander, L2 - Estimates and Existence Theorems for the $$\delta$$ Operator, Acta Math. 113 (1-2) pp 89– (1965) [6] Kerzman, Hölder and Lp - Estimates for Solutions of $$\delta$$u=f, Comm., Pure and Appl. Math. 24 (3) pp 301– (1971) [7] Kytmanov, The Bochner-Martinelli Integral and Its Applications (1992) · Zbl 0756.32001 [8] Lions, Problems aux Limites non Homogenes at Applications. I-III (1968) [9] Rempel, Index Theory of Elliptic Boundary Problems (1982) [10] Romanov., Spectral Analysis of Martinelli - Bochner Integral for Ball in Cn and Its Application, Functional, Anal, i Prilozhen. 12 (3) (1978) [11] Functional Anal. Appl. 12 3 1978 [12] Romanov, Convergence of Iterations of Bochner-Martinelli Operator, and the Cauchy Riemann System, Dokl. AN SSSR 242 (4) pp 780– (1978) · Zbl 0434.35066 [13] Soviet Math. Dokl. 19 5 1978 [14] Samborskii, Coercive Boundary - Values Problems for Overdetermined Systems (Elliptic Problems), Ukrainian Math. Journ. 36 (3) (1984) [15] Shimakura, Partial Differential Operators of Elliptic Type (1992) · Zbl 0757.35015 [16] Shlapunov, On Iterations of Green Type Integrals for Matrix Factorizations of the Laplace Operator, Rend. Mat. Ace. Line, s. 9 4 (5) (1994) · Zbl 0873.32016 [17] Shlapunov, Bases with Double Orthogonality in the Cauchy Problem for Systems with Injective Symbols, Proc. London Math. Soc. 71 (3) pp 1– (1995) · Zbl 0828.35040 [18] Tarkhanov, Parametrix Method in Theory of Differential Complexes (1990) [19] Tarkhanov , N. N. 1991
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.