Applications of the Differential Calculus to Nonlinear Elliptic Operators with Discontinuous Coefficients |
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Authors: | Dian K Palagachev Lutz Recke Lubomira G Softova |
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Institution: | 1. Dipartimento di Matematica, Politecnico di Bari, Via E. Orabona, 4, Bari, 70 125, Italy 2. Institut für Mathematik, Humboldt-Universit?t zu Berlin, Unter den Linden 6, Berlin, 10099, Germany 3. Bulgarian Academy of Sciences, Institute of Mathematics and Informatics, Sofia, Bulgaria
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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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Keywords: | 35J65 35R05 58C15 |
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