*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
\begin{equation}
\min\,\left\{
\int_{\Omega} f\left(u\,,\nabla u\right)\,dx\,\colon\,
u\in u_{0+} W^{{1,p}}(\Omega)
\right\}
\tag{$\mathcal{P}$}
\end{equation}
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.

**Keywords:**
existence of minimizers, convex integration, nonconvex variational problems

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