# A sharp attainment result for nonconvex variational problems

created on 23 May 2003
modified on 21 Jun 2004

[BibTeX]

Published Paper

Inserted: 23 may 2003
Last Updated: 21 jun 2004

Journal: Calc. Var.
Volume: 20
Pages: 301-328
Year: 2004

Abstract:

We consider the problem of minimizing autonomous, multiple integrals like \min\,\left\{ \int\Omega f\left(u\,,\nabla u\right)\,dx\,\colon\, u\in u0+ W{1,p}(\Omega) \right\} \tag{$\mathcal{P}$} where $f\colon\real\!\times\!{\real}^N\to [0\,,\infty)$ is a continuous, possibly nonconvex function of the gradient variable $\nabla u$. Assuming that the bipolar function $f^{\ast\ast}$ of $f$ is affine as a function of the gradient $\nabla u$ on each connected component of the sections of the detachment set $\mathcal{D}=\left\{f^{\ast\ast}<f\right\}$, we prove attainment for ($\mathcal{P}$) under mild assumptions on $f$ and $f^{\ast\ast}$. We present examples that show that the hypotheses on $f$ and $f^{\ast\ast}$ considered here for attainment are essentially sharp.