On the $\Gamma$-limit of singular perturbation problems with optimal profiles which are not one-dimensional. Part III: The energies with non local terms

TL;DR

This paper extends the analysis of $ ext{Gamma}$-limits for singular perturbation problems to include non-local terms, applying advanced techniques to problems in micromagnetics and conservation laws.

Contribution

It develops a method to establish upper and lower bounds for non-local variational problems, broadening the scope of $ ext{Gamma}$-convergence analysis beyond local energies.

Findings

Established bounds for non-local energy functionals

Applied techniques to micromagnetics and conservation laws

Reduced complex problems to previously studied frameworks

Abstract

We use the technique developed in [32]-[33] to construct the upper and the lower bounds for classes of problems containing non-local terms, including problems in micromagnetics and problems arising in the variational study of the Method of Vanishing Viscosity for systems of conservation laws. We reduced these problems to the problems considered in [32]-[33], with the appropriate prescribed differential constraint.