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

TL;DR

The paper proves that for certain coercive functionals on Banach spaces, the number of connected components of sublevel sets guarantees multiple local minima of perturbed functionals.

Contribution

It establishes a link between the topology of sublevel sets and the existence of multiple local minima under perturbations for coercive functionals.

Findings

Multiple local minima correspond to connected components of sublevel sets.

Weak topology connectedness influences the number of minima.

Perturbations preserve the number of minima under certain conditions.

Abstract

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 $\Psi^{-1}(]-\infty, r[)$.