This article discusses the general boundary value
problem for the nonlinear uniformly elliptic equation of second order
in D (0.1) and the boundary
condition
,(0.2) in a multiply
connected infinite domain D with the boundary T. The above boundary value problem is called Problem G.
Problem G extends the work [8] in which the equation (0.1) includes a nonlinear
lower term and the boundary condition (0.2) is more general. If the complex
equation (0.1) and the boundary condition (0.2) meet certain assumptions, some
solvability results for Problem G can be obtained. By using reduction to
absurdity, we first discuss a priori estimates of solutions and solvability for
a modified problem. Then we present results on solvability of Problem G.
[1] I. N. Vekua, “Generalized Analytic Functions,” Pergamon, Oxford, 1962.
[2] G. C. Wen, “Linear and Nonlinear Elliptic Complex Equations,” Shanghai Scientific and Technical Publishers, Shanghai, 1986. (in Chinese)
[3] G. C. Wen, “Conformal Mappings and Boundary Value Problems,” American Mathematical Society, Providence, 1992.
[4] G. C. Wen, “Approximate Methods and Numerical Analysis for Elliptic Complex Equations,” Gordon and Breach, Amsterdam, 1999.
[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, “Recent Progress in Theory and Applications of Modern Complex Analysis,” Science Press, Beijing, 2010.
[7] G. C. Wen and C. C. Yang, “On General Boundary Value Problems for Nonlinear Elliptic Equations of Second Order in a Multiply Connected Domain,” Acta Applicandae Mathematicae, Vol. 43 No. 2, 1996, pp. 169-189. doi:10.1007/BF00047923
[8] G. C. Wen, “Irregular Oblique Derivative Problems for Second Order Nonlinear Elliptic Equations on Infinite Domains,” Electronic Journal of Differential Equations, Vol. 2012, No. 142, 2012, pp. 1-8.