Sublevel sets and global minima of coercive functionals and local minima of their perturbations

Details Sublevel sets and global minima of coercive functionals and local minima of their perturbations Sublevel sets and global minima of coercive functionals and local minima of their perturbations

2004-02-26

arxiv.org/abs/math/0402444v3

Mathematics

Optimization and Control

12 pages

Scientific paper

The aim of the present paper is essentially to prove that if $\Phi$ and $\Psi$ are two sequentially weakly lower semicontinuous functionals on a reflexive real Banach space and if $\Psi$ is also continuous and coercive, then then following conclusion holds: if, for some $r > \inf_X \Psi$, the weak closure of the set $\Psi^{-1}(]-\infty, r[)$ has at least $k$ connected components in the weak topology, then, for each $\lambda > 0$ small enough, the functional $\Psi + \lambda\Phi$ has at least $k$ local minima lying in $\Psi^{-1}(]-\infty, r[)$.

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