Entropy method for line-energies

The aim of this talk is to study energy functionals concentrated on the jump set of 2D vector fields of unit length and of vanishing divergence. The motivation comes from thin-film micromagnetics where these functionals correspond to limiting wall-energies. The main issue consists in characterizing the wall-energy density (the cost function) so that the energy functional is lower semicontinuous (l.s.c.) in a certain space. The key point resides in the concept of entropies due to the scalar conservation law implied by our vector fields. Our main result identifies appropriate cost functions associated to certain sets of entropies. In particular, certain power cost functions lead to l.s.c. energy functionals. A second issue concerns the existence of minimizers of such energy functionals that we prove via a compactness result. A natural question is whether the viscosity solution is a minimizing configuration. We show that in general it is not the case for nonconvex domains. However, the case of convex domains is still open.

The talk is a joint work with Benoit Merlet, Ecole Polytechnique (Paris).