On the variational principle

摘要: 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> 0, there exists some point u ϵ, where∥ F′(u ϵ)∥∗⩽ ϵ, ie, its derivative can be made arbitrarily small. Applications are given to Plateau's problem, to partial differential equations, to nonlinear eigenvalues, to geodesics …