Résumé en anglais

The variational principle states that if a differentiable functional F attains its minimum at some point View the MathML source, then View the MathML source; 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 > 0, there exists some point uϵ, where View the MathML source, 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.