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A Morse lemma for degenerate critical points with low differentiability. (English) Zbl 1017.58007
Let $$f:U\subset\mathbb{H} \to\mathbb{R}$$ be a $$C^1$$ function defined on an open set $$U$$ of a Hilbert space $$\mathbb{H}$$. If $$f$$ is twice differentiable at 0 and $$A:\mathbb{H} \to\mathbb{H}$$ the symmetric operator defined by $$\langle Av,u\rangle =\frac 12 d^2f_0 (u,v)$$ then $$\mathbb{H}=N^\perp \oplus N$$, where $$N=\text{Ker}(A)$$.
One proves the following theorem: If $$f'$$ is strongly differentiable at the origin, there is a neighborhood $$V$$ of 0 in $$\mathbb{H}$$ and a homeomorphism $$\varphi :V\to\varphi (V)\subset\mathbb{H}$$ such that $f\bigl(\varphi (x,y)\bigr)= \frac 12 \langle Ax,x\rangle +f\bigl(g(y), y\bigr),\;d\varphi_0 =I,$ where $$g$$ is a function $$g:V\cap \mathbb{N}\to \mathbb{N}^\perp$$.
Reviewer: A. Neagu (Iaşi)

##### MSC:
 5.8e+06 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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