The Direct Method of the Calculus of Variations
The Direct Method of the Calculus of Variations
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变分法的直接法
DOI：
10.1007/978-1-4757-2061-7_1
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发表时间：
1989
期刊：
影响因子：
--
通讯作者：
M. Willem
中科院分区：
文献类型：
--
作者：
J. Mawhin;M. Willem
实变量的实函数φ在实数线上有界，它不需要有最小值，这从指数函数的例子中可以清楚地看出。如果我们将φ任意数列（ak）称为最小化数列（ak），使得φ具有收敛于a（take =a）的最小数列是实数的一个必要条件。如φ(x) =|x|定义的函数φ对于x≠0且φ(0) = 1的例子所示，虽然其所有的极小序列都收敛于0，但不能达到其最小值0。为了使收敛最小序列的极限a满足φ(a)= infφ，我们必须施加它
A real functionφof a real variable which is bounded below on the real line needs not to have a minimum, as it is clear from the example of the exponential function. If we callminimizing sequenceforφany sequence (ak) such thatask→ ∞, a necessary condition for the real numberato be such thatis thatφhas a minimizing sequence which converges toa(takeak=afor all integersk). Without suitable continuity assumptions onφthis condition will not be sufficient, as shown by the example of the functionφdefined byφ(x) =|x| forx≠ 0 andφ(0) = 1, which does not achieve its infimum 0 although all its minimizing sequences converge to zero. In order that the limit a of a convergent minimizing sequence be such thatφ(a)= infφ, we have to impose that