In [1], I. N. Vekua propose the Poincaré problem for some second order elliptic equations, but it can not be solved. In [2], the authors discussed the boundary value problem for nonlinear elliptic equations of second order in some bounded domains. In this article, the Poincaré boundary value problem for general nonlinear elliptic equations of second order in unbounded multiply connected domains have been completely investigated. We first provide the formulation of the above boundary value problem and corresponding modified well posed-ness. Next we obtain the representation theorem and a priori estimates of solutions for the modified problem. Finally by the above estimates of solutions and the Schauder fixed-point theorem, the solvability results of the above Poincaré problem for the nonlinear elliptic equations of second order can be obtained. The above problem possesses many applications in mechanics and physics and so on.
[1] I. N. Vekua, “Generalized Analytic Functions,” Pergamon, Oxford, 1962.
[2] G. C. Wen and H. Begehr, “Boundary Value Problems for Elliptic Equations and Systems,” Longman Scientific and Technical Company, Harlow, 1990.
[3] A. V. Bitsadze, “Some Classes of Partial Differential Equations,” Gordon and Breach, New York, 1988.
[4] G. C. Wen, “Conformal Mappings and Boundary Value Problems,” Translations of Mathematics Monographs 106, American Mathematical Society, Providence, 1992.
[5] G. C. Wen, D. C. Chen and Z. L. Xu, “Nonlinear Complex Analysis and Its Applications,” Mathematics Monograph Series 12, Science Press, Beijing, 2008.
[6] G. C. Wen, “Approximate Methods and Numerical Analysis for Elliptic Complex Equations,” Gordon and Breach, Amsterdam, 1999.
[7] G. C. Wen, “Linear and Quasilinear Complex Equations of Hyperbolic and Mixed Type,” Taylor & Francis, London, 2002. doi:10.4324/9780203166581
[8] G. C. Wen, “Recent Progress in Theory and Applications of Modern Complex Analysis,” Science Press, Beijing, 2010.