Applications of Differential Calculus to Nonlinear Elliptic Boundary Value Problems with Discontinuous Coefficients

Dian Palagachev , Lutz Recke , Lubomira Softova

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2004-21, 16p

MSC 2000

35J65 Nonlinear boundary value problems for linear elliptic PDE; boundary value problems for nonlinear elliptic PDE

35R05 PDE with discontinuous coefficients or data

58C15 Implicit function theorems; global Newton methods

Abstract
We deal with Dirichlet's problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. First we state suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Then we fix a solution $u_0$ such that the linearized in $u_0$ problem is non-degenerate, and we apply the Implicit Function Theorem: For all small perturbations of the coefficient functions there exists exactly one solution $u \approx u_0,$ and $u$ 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 coefficient functions can be general $L^\infty$-functions with respect to the space variable $x$. Moreover we show that the Newton Iteration Procedure can be applied to calculate a sequence of approximate (in $W^{2,p}$ again) solutions for $u_0.$