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

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 $φ, ψ:Y\to {\bf R}$ be two sequentially weakly lower semicontinuous functionals such that $$\inf_{y\in Y}{\min\{φ(y),ψ(y)\}\over {1+\|y\|^p}}>-\infty$$ 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 $γ\in L^{\infty}(T)\cap L^1(T)\setminus \{0\}$, with $γ\geq 0$, and for each $r>\inf_{Y}ψ$, if we put $$V_{γ,r}= \left \{u\in L^p(T,Y) : \int_Tγ(t)ψ(u(t))dμ\leq r\int_Tγ(t)dμ\right \}\ ,$$ we have $$\inf_{u\in V_{γ,r}} \int_Tγ(t)φ(u(t))dμ= \inf_{ψ^{-1}(r)}φ\int_Tγ(t)dμ .$$