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Shahrokhi-Dehkordi, M. S.
Traceless maps as the singular minimizers in the multi-dimensional 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 Euler-Lagrange 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, null-Lagrangian
Categories:49K27, 49N60, 49J30, 49K20

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