Schauder’s Technique of Proof for Existence Problems in Elliptic Differential Equations

Abstract

This technique is exceedingly well suited to settle existence questions for boundary-value problems of elliptic differential equations under very weak hypotheses on the coefficients. Such techniques are necessary if we want to treat the existence problems for quasi-linear or still more general equations. The technique of Schauder is based on the use of exceedingly sharp a priori estimates for the solution.⁴ In Section III-1.4 we met the simplest such estimate, with which, however, not very much can be done. We outline the method in a presentation which is not too general and without giving the complete hypotheses. However, the precise proof will not be any simpler than the one given in Section IV-3.