希尔伯特空间上非光滑泛函的分裂引理
The splitting lemmas for nonsmooth functionals on Hilbert spaces
摘要：
希尔伯特空间上退化临界点附近的Gromoll-Meyer广义Morse引理，即分裂引理，至少对$C^2$ -光滑泛函成立.本文对Hilbert空间$H$上的一类连续方向可微泛函（低于$C^1$）建立了一个分裂定理和一个移位定理，这类泛函在稠密连续嵌入Banach空间$X\subset H$上的临界点附近具有较高的光滑性（但低于$C^2$）. (This分裂定理概括了我所知的几乎所有以前的分裂定理）。此外，还给出了一个新的Poincaré-Hopf型定理以及$H$和$X$上泛函的临界群之间的关系.在临界子流形上给出了相应的形式。我们还将\cite{BaLi}中的Bartsch-Li无穷远处的分裂引理及其某些变形推广到希尔伯特空间上的一类连续方向可微泛函。我们的证明方法是将Duc-Hung-Khai \cite{DHK}中Morse-Palais引理的证明思想与\cite{JM, Skr, Va1}中的一些技巧相结合。我们的理论适用于紧致流形上的Lagrange系统和一大类非线性高阶椭圆型方程的边值问题。
英文摘要：
The usual Gromoll-Meyer's generalized Morse lemma near degenerate critical points on Hilbert spaces, so called splitting lemma, is stated for at least $C^2$-smooth functionals. In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower than $C^1$) on a Hilbert space $H$ which have higher smoothness (but lower than $C^2$) on a densely and continuously imbedded Banach space $X\subset H$ near a critical point lying in $X$. (This splitting theorem generalize almost all previous ones to my knowledge). Moreover, a new theorem of Poincaré-Hopf type and a relation between critical groups of the functional on $H$ and $X$ are given. The corresponding version at critical submanifolds is presented. We also generalize the Bartsch-Li's splitting lemma at infinity in \cite{BaLi} and some variants of it to a class of continuously directional differentiable functionals on Hilbert spaces. Our proof methods are to combine the proof ideas of the Morse-Palais lemma due to Duc-Hung-Khai \cite{DHK} with some techniques from \cite{JM, Skr, Va1}. Our theory is applicable to the Lagrangian system on compact manifolds and boundary value problems for a large class of nonlinear higher order elliptic equations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
