Topological Fixed Point Theorems and Their Application

In this chapter the solvability of the classical Dirichlet problem for quasilinear equations is reduced to the establishment of certain apriori estimates for solutions. This reduction is achieved through the application of topological fixed point theorems in appropriate function spaces. We shall first formulate a general criterion for solvability and illustrate its application in a situation where the required apriori estimates are readily derived from our previous results. The derivation of these apriori estimates under more general hypotheses will be the major concern of the ensuing chapters.