Abstract view
Traceless Maps as the Singular Minimizers in the Multidimensional Calculus of Variations


Published:20170221
Printed: Sep 2017
M. S. ShahrokhiDehkordi,
Department of Mathematics, University of Shahid Beheshti, Evin, Tehran, Iran
Abstract
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.