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Applications of the differential calculus to nonlinear elliptic operators with discontinuous coefficients. (English) Zbl 1194.35157

Summary: 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 \(u \approx u_0\) which 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 coefficients can be general \(L^{\infty}\)-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 \(W^{2, p}\)) solutions for \(u_0\).

MSC:

35J60 Nonlinear elliptic equations
35A35 Theoretical approximation in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
58C15 Implicit function theorems; global Newton methods on manifolds
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