Integral functionals on L^p-spaces: infima over sub-level sets

2013/12/19 by Biagio Ricceri, Ricceri, Biagio
Computer Science · Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.1312.5715

openalex publication_date 2013/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we establish the following result: Let (T,\cal F,μ) be a σ-finite measure space, let Y be a reflexive real Banach space, and let φ, ψ:Y→ \bf R be two sequentially weakly lower semicontinuous functionals such that inf_{y∈ Y}min\φ(y),ψ(y)\\over 1+‖y‖^pgt;-∞ for some p>0. Moreover, assume that φ has no global minima, while φ+λψ is coercive and has a unique global minimum for each λ>0. Then, for each γ∈ L^∞(T)∩ L¹(T)∖ \0\, with γ≥ 0, and for each r>inf_Yψ, if we put V_γ,r= \u∈ L^p(T,Y) : ∫_Tγ(t)ψ(u(t))dμ≤ r∫_Tγ(t)dμ \ , we have inf_u∈ V_γ,r ∫_Tγ(t)φ(u(t))dμ= inf_ψ^-1(r)φ∫_Tγ(t)dμ .

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