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

  Published:2017-02-21
 Printed: Sep 2017
  • M. S. Shahrokhi-Dehkordi,
    Department of Mathematics, University of Shahid Beheshti, Evin, Tehran, Iran
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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 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 traceless map, singular minimizer, null-Lagrangian
MSC Classifications: 49K27, 49N60, 49J30, 49K20 show english descriptions Problems in abstract spaces [See also 90C48, 93C25]
Regularity of solutions
Optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
Problems involving partial differential equations
49K27 - Problems in abstract spaces [See also 90C48, 93C25]
49N60 - Regularity of solutions
49J30 - Optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49K20 - Problems involving partial differential equations
 

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