Local Minimizers of Integral Functionals are Global Minimizers

Abstract

We show that local minimizers of integral functionals associated with a measurable integrand $f:\Omega \times E \to \mathbb {R} \cup \{ \pm \infty \}$ are actually global minimizers. Here $(\Omega , \mathcal {S},\mu )$ is a measured space with an atomless $\sigma$-finite positive measure, E is a separable Banach space, and the integral functional ${I_f}(x) = \smallint _\Omega ^ \ast f(\omega ,x(\omega ))d\mu$ is defined on ${L_p}(\Omega ,E)$ or, more generally, on some decomposable set of measurable mappings x from $\Omega$ into E.