Cartesian currents in the calculus of variations 2

Non-scalar variational problems arise in various fields, including geometry, where they relate to harmonic maps between Riemannian manifolds and minimal immersions. These issues also appear in physics, particularly in classical a-models. In continuum mechanics, non-linear elasticity serves as another example, while the Oseen-Frank theory of liquid crystals and Ginzburg-Landau theory of superconductivity necessitate the treatment of variational problems to model complex phenomena. The primary focus is often on identifying energy-minimizing representatives within homology or homotopy classes of maps, as well as minimizers with specific topological singularities, topological charges, and stable deformations. Over the past few decades, there has been an increasing interest and understanding of the general theory surrounding these geometric variational problems. However, the absence of a regularity theory in the non-scalar case, in contrast to the scalar case, complicates matters. This is due to the presence of singularities in vector-valued minimizers, which are often linked to concentration phenomena in energy density. Consequently, discerning a weak formulation or fully grasping the implications of various weak formulations becomes a complex task.