广义函数的经典变分理论
The classical theory of calculus of variations for generalized functions
摘要：
本文将经典变分理论推广到广义函数。该框架是广义光滑函数的范畴，它包括Schwartz分布，同时与普通光滑函数共享许多非线性性质。我们证明了极值与Euler-Lagrange方程之间的完全联系，经典的极小值存在的充要条件，Legendre必要条件，Jacobi关于共轭点的定理和Noether定理.我们关闭低正则黎曼几何的应用。
英文摘要：
We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler-Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi's theorem on conjugate points and Noether's theorem. We close with an application to low regularity Riemannian geometry.
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