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

Shahrokhi-Dehkordi, M. S.

Canadian mathematical bulletin, Tome 60 (2017) no. 3, pp. 631-640

Résumé

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.

Détail

DOI : 10.4153/CMB-2016-094-5

Mots-clés : 49K27, 49N60, 49J30, 49K20, traceless map, singular minimizer, null-Lagrangian