Gráficos com curvatura média prescrita

Abstract

In this work, we shall study the problem of existence of graphs with prescribed mean curvature in euclidean spaces. Analytically the problem consists of a quasilinear elliptic equation with Dirichlet condition. To keep the presentation self-contained, we show the relevant results of the theory or linear and quasilinear elliptic partial differential equations. For linear equations, we prove the Schauder estimate for classical solutions, and apply the continuity method to obtain an existence theorem for solutions of the linear Dirichlet problem. For quasilinear equations, we prove a version of the comparison principle, and use the fi xed point method to show that solvability of quasilinear equations is connected to the existence of appropriate a priori estimates for solutions. Armed with these results, we establish appropriate hypothesis for the demonstration of a priori estimates for solutions of the equation of graphs with prescribed mean curvature, which leads to the fi nal result of the work.