On the variational principle

Abstract

The variational principle states that if a differentiable functional F attains its minimum at some point u ̄, then F′( u ̄) = 0; it has proved a valuable tool for studying partial differential equations. This paper shows that if a differentiable function F has a finite lower bound (although it need not attain it), then, for every ε{lunate} > 0, there exists some point uε{lunate}, where ∥F′(uε{lunate})∥* ≤ ε{lunate}, i.e., its derivative can be made arbitrarily small. Applications are given to Plateau's problem, to partial differential equations, to nonlinear eigenvalues, to geodesics on infinite-dimensional manifolds, and to control theory. © 1974.