Applications of Differential Calculus to Nonlinear Elliptic Boundary Value Problems with Discontinuous Coefficients

We deal with Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. First we state suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Then we fix a solution u0 such that the linearized in u0 problem is non-degenerate, and we apply the Implicit Function Theorem: For all small perturbations of the coefficient functions there exists exactly one solution u ≈ u0, and u depends smoothly (in W 2,p with p larger than the space dimension) on the data. For that no structure and growth conditions are needed, and the perturbations of the coefficient functions can be general L∞-functions with respect to the space variable x. Moreover we show that the Newton Iteration Procedure can be applied to calculate a sequence of approximate (in W 2,p again) solutions for u0.