Applications of the differential calculus to nonlinear elliptic operators with discontinuous coefficients

The paper concerns Dirichlet's problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u(0) such that the linearized at u(0) problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u approximate to u(0) which depends smoothly (in W-2,W-p supercript stop with p larger than the space dimension) on the data. For that, no structure and growth conditions are needed and the perturbations of the coefficients can be general L-infinity-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W-2,W-p) solutions for u (0).