Calculus of Variations and Geometric Measure Theory

F. Maggi

On the relaxation on $BV$ of certain non coercive integral functionals

created on 14 Dec 2002
modified by maggi on 19 Dec 2005


Published Paper

Inserted: 14 dec 2002
Last Updated: 19 dec 2005

Journal: J. Convex Anal.
Volume: 10
Number: 2
Pages: 477-489
Year: 2003


In this paper we prove an integral representation formula for the relaxed functional of a scalar non parametric integral of the Calculus of Variations. Similar results are known to be true under the key assumption that the integrand is coercive in the gradient variable 1. Here we show that the same integral representation holds for a wide class of non coercive integrands, including for example the strictly convex ones.


The weak coercivity property under consideration, known as \textit{demicoercivity}, was introduced in 2 and recently connected to the study of Serrin's lower semicontinuity theorem 3.


1 Dal Maso G., \textit{Integral representation on $BV(\Omega$) of $\Gamma$-limits of variational integrals}, Manuscripta Math. \textbf{30}, 1980, 387-416.

2 Anzellotti G., Buttazzo G., Dal Maso G., \textit{Dirichlet problem for demicoercive functionals}, Nonlinear Anal. \textbf{10}, 1986, 603-613.

3 Gori M., Maggi F., \textit{The common root of the geometric conditions in Serrin's lower semicontinuity theorem}, preprint Dip. Mat. U. Dini, Firenze, 2002.

Keywords: relaxation, calculus of variations, demicoercivity, functions of bounded variation