Local existence theorem for the solution of the generalized Laplace condition
The generalized Laplace condition describing equilibrium surface of pendant drop with intermediate layer is considered and the corresponding Cauchy problem is formulated. The main part of the generalized Laplace condition is not linear. We construct non-linear differential operator of the first order and formulate Cauchy problem for it. Using Shauder theorem we prove that it has analytic positive solution. We prove that the coefficients of the series representing this solution can be used as upper bounds for the moduli of the coefficients of the series formally representing a solution of the generalized Laplace condition. Thus, the existence of analytic solution of nonlinear equation representing generalized Laplace condition is proved. The theorem we proved gives a possibility to calculate with any degree of approximation the form of the drop. The method without any serious alterations can be used in order to investigate sessile drops.