The index of a constrained critical point. (English) Zbl 0803.58012

Hassell, Catherine; Rees, Elmer

Am. Math. Mon. 100, No. 8, 772-778 (1993).

Let \(M^ n \subset \mathbb{R}^{n + m}\) be a smooth submanifold described as the common zero-set of \(m\) smooth real-valued functions \(g_ 1,\dots,g_ m\) on \(\mathbb{R}^{n + m}\) and let \(f\) be a smooth function on \(\mathbb{R}^{n + m}\). Defining \(L(x,\lambda) = f(x) + \lambda_ 1 g_ 1(x) + \dots + \lambda_ m g_ m(x)\) for \(x \in \mathbb{R}^{n + m}\) and \(\lambda = (\lambda_ 1, \dots,\lambda_ m) \in \mathbb{R}^ m\), the authors compare the Hessians of \(f \mid_ M\) and \(L\) for obtaining informations on the nature of critical points of \(f\) on \(M\). There are given specific examples.

Reviewer: D.Motreanu (Iaşi)

MSC:

58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E50	Applications of variational problems in infinite-dimensional spaces to the sciences

Keywords:

Lagrange multiplier; Hessian; critical points