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1991 | 54 | 1 | 45-57
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An integral formula for the derivatives of solutions of certain elliptic systems

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 Abstract. As was shown by Polozhii and Shabat, the solutions ƒ of elliptic systems $ƒ = νƒ_{\bar{z}} + μ{\bar{ƒ_z}$ satisfy a generalized Cauchy integral formula. Here we will show that the derivatives fz, too, satisfy an integral formula. This formula, announced already in [6], rests upon the notion of generalized (— l)th powers and represents an astonishing analogue to the classical Cauchy integral formula for the derivative of an analytic function. The proof rests on an integral relation for generalized powers, which is of independent interest and which likewise represents a generalization of a classical relation.
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  • Martin-Luther-Universität Halle-Wittenberg, Sektion Mathematik, 0-4010 Halle/S., Germany
Bibliografia
  • [1] L. Bers, Partial differential equations and generalized analytic functions, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 130-136; Second note, ibid. 37 (1951), 42-47.
  • [2] B. Bojarski, Generalized solutions of a system of differential equations of first order and elliptic type with discontinuous coefficients. Mat. Sb. (N.S.) 43 (85) (1957), 451 503 (in Russian).
  • [3] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, 2nd ed,, Springer, Berlin 1973.
  • [4] G. N. Polozhii, A generalization of Cauchy's integral formula, Mat. Sb. (N.S.) 24 (66) (1949), 375-384 (in Russian).
  • [5] H. Renelt, Elliptic Systems, and Quasiconformal Mappings, Wiley, 1988.
  • [6] H. Renelt, Generalized powers in the theory of (ν, μ)-solutions, Ann. Univ. Mariae Curie-Sklodowska Sect. A 40 (1986), 217-235.
  • [7] B. V. Shabat, Cauchy's theorem and formula for quasiconformal mappings of linear classes, Dokl. Akad, Nauk SSSR (N.S.) 69 (1949), 305-308 (in Russian).
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Bibliografia
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