Elliptic Partial Differential Equations

Abstract

This chapter describes an approximation theory of quadratic forms that is applicable to linear-elliptic multiple integral problems. It discusses the theory of quadratic forms by Dennemeyer. The exposition is intended to parallel the earlier development and in particular the second-order problems. Dennemeyer's ideas are contained in his dissertation and research article. They follow from ideas in Hestenes. Reference in Dennemeyer's work contains many of the technical details for quadratic forms. These details include ellipticity, Gaarding's inequality, and coerciveness. The main results of approximation theory are described in the chapter in terms of inequalities involving nonnegative indices. In particular, it is illustrated in the chapter that the hypothesis for these inequalities is sufficiently general to include the resolution space or λ theory of focal point as well as the continuous perturbations of coefficients of quadratic forms and partial differential equations. The approximation setting is extended to obtain an approximate theory of conjugate surfaces. These results are then interpreted to obtain existence theorems and other properties for the multiple integral problems. The chapter also discusses comparison theorems for quadratic forms and partial differential equations. © 1980, Academic Press Inc.