一些C^2泛函的临界点定理的非光滑推广

Nonsmooth generalization of some critical point theorems for C^2 functionals

本文基于裂开定理的新近结果,并结合度量临界点理论与局部Lipsitich泛函的临界点理论,推广经Morse理论方法获得的一些对C^2泛函的临界点定理到一类Frchet可微且连续方向可微泛函.一个关键是,对Banach空间开集上的Frchet可微且严格Hadamard可微(比局部Lipschitz连续强但比连续方向可微弱)的泛函,观察到它作为连续泛函的度量临界点集、作为Frchet可微泛函的临界点集及作为局部Lipschitz连续泛函的临界点集都一致.

In this paper, we combine the author＇s recent results on splitting theorems, with metric critical point theory and critical point theory for locally Lipschitz functionals, to generalize some critical point theorems for C^2 functionals obtained via Morse theoretic methods to a class of both Frchet differentiable and continuously directional differential functionals. As a key, if a functional on an open subset of a Banach space is Frchet differentiable, and strictly Hadamard differentiable（which is stronger than locally Lipschitz continuity, but weaker than continuously directional differentiability）, we observe that it has the same critical set as a continuous functional on a metric space, a Fr′echet differentiable one, and a locally Lipschitz continuous one.