Entropy method for line-energies

In this paper we analyze energy functionals concentrated on the discontinuity lines of unitlength, divergence-free 2D vector fields. The motivation comes from thin-film micromagnetics where these functionals correspond to mesoscopic wall-energies. A natural issue consists in characterizing the line-energy densities for which the functionals are lower semicontinuous for a relevant topology. In fact, this is a necessary condition for being the Γ−limit of a family of energies. A key point in our study is the use of the notion of entropy production borrowed from the field of conservation laws. With this tool, we build a large class of lower semicontinuous line-energies. In particular, we prove that certain power functions lead to such line-energy functionals as conjectured in [2]. We also deduce compactness properties for these functionals leading to the existence of minimizers for their lower semicontinuous envelopes. Another natural question is whether the viscosity solution is a minimizing configuration. We show that the answer is in general negative by exhibiting some special nonconvex domains as counterexamples. However, we establish positive results for some special domains (stadium, ellipse and union of two discs). The case of general convex domains is still open. AMS classification: 49J45 (49Q20, 58E20, 49L25, 35A15, 82D40)