*Published Paper*

**Inserted:** 20 apr 2005

**Last Updated:** 26 oct 2005

**Journal:** SIAM J. Control Optim.

**Volume:** 44

**Number:** 4

**Pages:** 1370-1390

**Year:** 2005

**Abstract:**

We study the existence
of Lipschitz minimizers of integral functionals
$$ \mathcal{I}(u)=\int_{{\Omega}
}
\varphi(x,\textrm{det}\,Du(x))\,dx$$
where $\Omega$ is an open subset of $\mathbb{R}^N$ with Lipschitz boundary,
$\varphi:\Omega\times (0,+\infty)\to [0,+\infty)$ is a continuous
function and $u\in W^{1,N}(\Omega, \mathbb{R}^N)$, $u(x)=x$ on
$\partial \Omega$.
We consider both
the cases of $\varphi$ convex and nonconvex
with respect to the last variable. The attainment results are obtained
passing through the minimization of an auxiliary functional
and the solution of a prescribed jacobian equation.

**Keywords:**
Lipschitz regularity, nonpolyconvex functional, existence of minimizers, prescribed jacobian equation

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