On the unification of the calculus of variations and the theory of monotone nonlinear operators in banach spaces.

The recently developed theory1 of monotone nonlinear operators T from a Banach space X to its conjugate space X* can be considered most naturally as an extension to nonvariational problems of the basic ideas of the direct method of the calculus of variations. First of all, in practice, its simplest and most basic application is to a class of nonlinear elliptic boundary value problems2 which is an extension of the class of Euler-Lagrange equations of multiple integral problems of general order, paralleling the application of Hilbert space methods for general linear elliptic problems as an extension of the variational method for self-adjoint problems. Second, on a more basic level of principle, if the operator T is the derivative (Gateaux or Frechet) of a real-valued function f on X, the condition that T be monotone is equivalent to the convexity off,3 the basic property (with its modifications) on which one rests the direct method of the calculus of variations in Banach spaces.4 For this subclass of monotone operators T, the existence of solutions of the equation Tu = 0 is equivalent to the existence of critical points for the corresponding functionf. On the other hand, for f just semiconvex, the direct method of the calculus of variations yields not merely critical points of f but extreme points of f, so that a certain amount of information is lost in passing from the study of variational problems to the theory of monotone operators. A similar relation holds between variational problems on convex sets and monotone operator inequalities on convex sets.5 It is our object in the present note to give a unified extension of the theory of monotone operator equations, the calculus of variations, and the theory of monotone operator inequalities on convex sets. The formulation of this extension is as follows: Let X be a reflexive Banach space, X* its conjugate space with the pairing between w in X* and u in X denoted by (wu). If T is a mapping from X to X* (or has its domain D(T) in X), T is said to be monotone if