Calculus of Variations and Geometric Measure Theory

N. Fusco - M. Gori - F. Maggi

A remark on Serrin's Theorem

created on 06 Nov 2003
modified by maggi on 19 Dec 2005


Accepted Paper

Inserted: 6 nov 2003
Last Updated: 19 dec 2005

Journal: Nonlinear Differential Equations and Applications
Year: 2003

Preprint n. 48-2003 Dipartimento di Matematica e Applicazioni "R. Caccioppoli"


The main theorem is: Let $\Sigma$ be an open set in $\mathbb{R}^d$ and $f=f(t,\xi):\Sigma\times\mathbb{R}^n\goto\mathbb{R}$ be a non negative and lower semicontinuous function with $f(t,\cdot)$ convex and demicoercive. Then there exists a sequence of functions $\{f_m\}_{m\in\nat}$ of class $C^\infty(\Sigma\times\mathbb{R}^n,[0,\infty))$ such that \[ f=\sup_{m\in\nat}f_m. \]

This statement allows to unify the three-parts statement of Serrin's lower semicontinuity theorem.

Keywords: demicoercivity, Lower Semicontinuity