Charles B. Morrey

Multiple Integral Peoblems in the Calculus of Variations and Related Topics

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.

Differentiation in ℝ N

There are two principal extensions to ℝ N of the theory of differentiation of real-valued functions on ℝ1. In this section, we develop the natural generalization of ordinary differentiation discussed in Chapter 4 to partial differentiation of functions from ℝ N to ℝ1. In Section 7.3, we extend the ordinary derivative to the total derivative.

Functions Defined by Integrals; Improper Integrals

The solutions of problems in differential equations, especially those which arise in physics and engineering, are frequently given in terms of integrals. Most often either the integrand of the integral representing the solution is unbounded or the domain of integration is an unbounded set. In this chapter we develop rules for deciding when it is possible to interchange the processes of differentiation and integration—commonly known as differentiation under the integral sign. When the integrand becomes infinite at one or more points or when the interval of integration is infinite, a study of the convergence of the integral is needed in order to determine whether or not the differentiation process is allowable. We establish the required theorems for bounded functions and domains in this section and treat the unbounded case in Sections 11.2 and 11.3.

Contraction Mappings, Newton’s Method, and Differential Equations

The main result of this section is a simple theorem which proves to be useful in the solution of algebraic and differential equations. It will also be used to prove the important Implicit function theorem in Chapter 14.

Implicit Function Theorems and Lagrange Multipliers

Suppose we are given a relation in ℝ2 of the form

Differentiation under the Integral Sign. Improper Integrals. The Gamma Function

F\left( {x,y} \right) = 0.

Vector Field Theory

(14.1) Then to each value of x there may correspond one or more values of y which satisfy (14.1)—or there may be no values of y which do so. If I = x: x0 − h < x < x0 + h is an interval such that for each x - I there is exactly one value of y satisfying (14.1), then we say that F(x, y) = 0 defines y as a function of x implicitly on I. Denoting this function by f, we have F[x, f(x)] = 0 for x on I.

Basic properties of functions on ℝ1

In Chapters 2 through 5 we developed many properties of functions from ℝ1 into ℝ1 with the purpose of proving the basic theorems in differential and integral calculus of one variable. The next step in analysis is the establishment of the basic facts needed in proving the theorems of calculus in two and more variables. One way would be to prove extensions of the theorems of Chapters 2–5 for functions from ℝ2 into ℝ1, then for functions from ℝ3 into ℝ1, and so forth. However, all these results can be encompassed in one general theory obtained by introducing the concept of a metric space and by considering functions defined on one metric space with range in a second metric space. In this chapter we introduce the fundamentals of this theory and in the following two chapters the results are applied to differentiation and integration in Euclidean space in any number of dimensions.