Solutions to quasilinear equations by an iterative method
Pablo Amster, M. M. Cassinelli, Maria Cristina Mariani · Bulletin of the Belgian Mathematical Society - Simon Stevin · 2000
We apply an iterative method in order to construct a solution to the mean curvature equation for nonparametric surfaces. INTRODUCTION The prescribed mean curvature equation with Dirichlet condition for a nonparametric surface X :\\Omega \\Gamma! IR 3 , U(x; y) = (x; y; u(x; y)) is the quasilinear partial differential equation (1) ( (1 + u 2 y )u xx + (1 + u 2 x )u yy \\Gamma 2u x u y u xy = 2h(u) i 1 + jruj 2 j 3 2 in\\Omega u = g in @\\Omega where \\Omega is a bounded domain in IR 2 , and h : IR \\Gamma! IR is a given continuous function. This problem and the general parametric case have been studied by several authors, see e.g. [2-5,6,7,9-13]. SOLUTIONS BY AN ITERATIVE METHOD We'll apply an iterative method inspired in the Newton Imbedding procedure [8]. For this purpose, let us define for each v 2 C 1 (\\Omega ) the bounded linear operator Q v : W 2;p ! L p (\\Omega\\Gamma given by Q v u = 1 2(1 +rv 2 ) 3 2 ((1 + v 2 y )u xx + (1 + v 2 x )u yy \\Gamma 2v x...