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).

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

Palagachev, Dian;Softova, Lyoubomira
2006-01-01

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 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).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/4701539
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