On the variational principle

The variational principle states that if a differentiable functional F attains its minimum at some point $\text{u}\text{?}$, then $\text{F′(}\text{u}\text{?}\text{) = 0}$; 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 ? > 0, there exists some point u_?, where $\text{∥F′(u}{}_{\mathrm{?}}\text{)∥}{}^1\text{? ?}$, 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.