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 Kostadinov; Recke, L.; Softova, L. G.. - In: MATHEMATISCHE ANNALEN. - ISSN 0025-5831. - 336:3(2006), pp. 617-637. [10.1007/s00208-006-0014-x]
Applications of the differential calculus to nonlinear elliptic operators with discontinuous coefficients
PALAGACHEV, Dian Kostadinov;
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).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.