arXiv subjects
Publications and source records attributed to Biagio Ricceri.
Perhaps, it is not too far from the truth to say that, among the great concepts (as compactness, completeness, order, convexity) on which functional analysis is based, connectedness is relatively less popular, though this does not mean that it is less useful than the others. The aim of this lecture is just to support this latter sentence, focusing some recent results where connectedness plays a central role.
In this paper, I propose some problems, of topological nature, on the energy functional associated to the Dirichlet problem -Δu = f(x,u) in Omega, u restricted to the boundary of Omega is 0. Positive answers to these problems would produce innovative multiplicity results on this Dirichlet problem.
The aim of the present paper is essentially to prove that if $Φ$ and $Ψ$ are two sequentially weakly lower semicontinuous functionals on a reflexive real Banach space and if $Ψ$ is also continuous and coercive, then then following conclusion holds: if, for some $r > \inf_X Ψ$, the weak closure of the set $Ψ^{-1}(]-\infty, r[)$ has at least $k$ connected components in the weak topology, then, for each $λ> 0$ small enough, the functional $Ψ+ λΦ$ has at least $k$ local minima lying in $Ψ^{-1}(]-\infty, r[)$.
In this paper, two multiplicity results about local minima of integrals of the calculus of variations are established. The main tool used to prove them is the theory developed in [B. Ricceri, Sublevel sets and global minima of coercive functionals and local minima of their pertubations, math.OC/0402444].
We prove, under certain conditions, the existence of zeros for a weakly continuous operator on a paracompact topological space into the dual of a Banach space.