VARIATIONAL PRINCIPLE
VARIATIONAL PRINCIPLE
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DOI：
10.1016/0022-247x(74)90025-0
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发表时间：
1974-01-01
影响因子：
1.3
通讯作者：
EKELAND, I
中科院分区：
文献类型：
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作者：
EKELAND, I
变分原理指出，如果一个可微泛函F在某一点ū处达到极小值，则F‘(ū)=0；它已被证明是研究偏微分方程的一个有价值的工具。本文证明了：如果一个可微函数F有一个有限的下界(虽然它不需要达到它)，则对每个ϵ>；0，存在某个点uϵ，其中∥F‘(uϵ)∥∗⩽ϵ，即它的导数可以使其任意小)。应用于高原问题、偏微分方程组、非线性特征值、无限维流形上的测地线和控制理论。
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 on infinite-dimensional manifolds, and to control theory.