Global strong solvability of Dirichlet problem for a class of nonlinear elliptic equations in the plane

Global solvability and uniqueness results are established for Dirichlet's problem for a class of nonlinear differential equations on a convex domain in the plane, where the nonlinear operator is elliptic in sense of Campanato. We prove existence by means of the Leray-Schauder fixed point theorem, using Alexandrov-Pucci maximum principle in order to find a priori estimate for the solution

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

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.