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

TL;DR

This paper investigates the infimum of integral functionals on $L^p$-spaces over sub-level sets, establishing conditions under which the infimum reduces to a simpler form involving the boundary of the sub-level set.

Contribution

It provides new conditions linking the properties of two functionals and the structure of sub-level sets to simplify the computation of infima in $L^p$-spaces.

Findings

Infimum over sub-level sets equals a boundary value of the functional.

Conditions on coercivity and minimality ensure the infimum simplifies.

Results apply to reflexive Banach space-valued functions.

Abstract

In this paper, we establish the following result: Let $(T,{\cal F},\mu)$ be a $\sigma$-finite measure space, let $Y$ be a reflexive real Banach space, and let $\varphi, \psi:Y\to {\bf R}$ be two sequentially weakly lower semicontinuous functionals such that $$\inf_{y\in Y}{{\min\{\varphi(y),\psi(y)\}}\over {1+\|y\|^p}}>-\infty$$ for some $p>0$. Moreover, assume that $\varphi$ has no global minima, while $\varphi+\lambda\psi$ is coercive and has a unique global minimum for each $\lambda>0$. Then, for each $\gamma\in L^{\infty}(T)\cap L^1(T)\setminus \{0\}$, with $\gamma\geq 0$, and for each $r>\inf_{Y}\psi$, if we put $$V_{\gamma,r}= \left \{u\in L^p(T,Y) : \int_T\gamma(t)\psi(u(t))d\mu\leq r\int_T\gamma(t)d\mu\right \}\ ,$$ we have $$\inf_{u\in V_{\gamma,r}} \int_T\gamma(t)\varphi(u(t))d\mu= \inf_{\psi^{-1}(r)}\varphi\int_T\gamma(t)d\mu\ .$$