Upper bounds for a class of energies containing a non-local term

Poliakovsky, Arkady

ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 4, pp. 856-886

Résumé

In this paper we construct upper bounds for families of functionals of the form

$$E_{\epsilon }\left(\phi \right):={\int }_{\Omega }\left({\epsilon \left|\nabla \phi \right|}^2+\frac{1}{\epsilon }W\left(\phi \right)\right)\mathrm{d}x+\frac{1}{\epsilon }{\int }_{{ℝ}^N}{\left|\nabla {\overline{H}}_{F\left(\phi \right)}\right|}^2\mathrm{d}x$$

where Δ ${\overline{H}}_u$ = div {${\chi }_{\Omega }$ u}. Particular cases of such functionals arise in Micromagnetics. We also use our technique to construct upper bounds for functionals that appear in a variational formulation of the method of vanishing viscosity for conservation laws.

Détail

DOI : 10.1051/cocv/2009022

Classification : 35A15, 35J35, 82D40
Keywords: gamma-convergence, micromagnetics, non-local energy