A Morse lemma for degenerate critical points with low differentiability. (English) Zbl 1017.58007

Moura, Adriano A. de; de Souza, Fausto M.

Abstr. Appl. Anal. 5, No. 2, 113-118 (2000).

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:

58E05

Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Keywords:

twice strongly differentiable function; degenerate critical point; Hilbert space; symmetric operator