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( Problems involving partial differential equations )
1. CMB Online first
 ShahrokhiDehkordi, M. S.

Traceless maps as the singular minimizers in the multidimensional calculus of variations
Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain and
consider
the energy functional
\begin{equation*}
{\mathcal F}[u, \Omega] := \int_{\Omega} {\rm F}(\nabla {\bf u}(\bf x))\, d{\bf x},
\end{equation*}
over the space of $W^{1,2}(\Omega, \mathbb{R}^m)$ where the integrand
${\rm F}: \mathbb M_{m\times n}\to \mathbb{R}$ is a smooth uniformly
convex function
with bounded second derivatives. In this paper we address the
question of
regularity for solutions of the corresponding system of
EulerLagrange equations.
In particular we introduce a class of singular maps referred
to as traceless and
examine them as a new counterexample to the regularity of minimizers
of the energy
functional $\mathcal F[\cdot,\Omega]$ using a method based on
null Lagrangians.
Keywords:traceless map, singular minimizer, nullLagrangian Categories:49K27, 49N60, 49J30, 49K20 
