Multiple Integral Peoblems in the Calculus of Variations and Related Topics

Abstract

In this series of lectures, I shall present a greatly simplified account of some of the research concerning multiple integral problems in the calculus of variations which has been reported in detail in the papers [39|, [40], [41], [42], [44], [46], and [47]. I shall speak only of problems in non-parametric form and shall therefore not describe the excellent result concerning double integrals in parametric form obtained almost concurrently by Sigalov, Danskin, and Cesari [62], [9|, [5]) nor the work of L.C. Young and others on generalized surfaces. Some of my results have been extended in various ways by Cinquini [6], De Giorgi [10], Fichera [17], Nobeling [51], Sigalov [58], [59], [60], [61], Silova [63], and Stainpacchia [67], [68], [69], [70]. However, it is hoped that the results presented here will serve as an introduction to the subject.