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Traceless Maps as the Singular Minimizers in the Multi-dimensional Calculus of Variations

Published online by Cambridge University Press:  20 November 2018

M. S. Shahrokhi-Dehkordi
M. S. Shahrokhi-Dehkordi*
Affiliation:
Department of Mathematics, University of Shahid Beheshti, Evin, Tehran, Iran. e-mail: m_shahrokhi@sbu.ac.ir
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Abstract

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Let $\Omega \subset {{\mathbb{R}}^{n}}$ be a bounded Lipschitz domain and consider the energy functional

1

$$F[u,\Omega ]\,:=\,\,{{\int }_{\Omega }}\text{F}(\triangledown u(x))\,d\text{x,}$$

over the space of ${{W}^{1,2}}(\Omega ,{{\mathbb{R}}^{m}})$ where the integrand $\text{F}:{{\mathbb{M}}_{m\times n}}\to \mathbb{R}$ 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 singularmaps referred to as traceless and examine themas a new counterexample to the regularity of minimizers of the energy functional $F[\cdot ,\Omega ]$ using a method based on null Lagrangians.

Keywords

49K2749N6049J3049K20traceless mapsingular minimizernull-Lagrangian
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 3 , 01 September 2017 , pp. 631 - 640
Copyright
Copyright © Canadian Mathematical Society 2017

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