Abstract
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 ≈ 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 ∞-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.
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Palagachev, D.K., Recke, L. & Softova, L.G. Applications of the Differential Calculus to Nonlinear Elliptic Operators with Discontinuous Coefficients. Math. Ann. 336, 617–637 (2006). https://doi.org/10.1007/s00208-006-0014-x
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DOI: https://doi.org/10.1007/s00208-006-0014-x