By Mariano Giaquinta
Non-scalar variational difficulties seem in several fields. In geometry, for in stance, we come across the fundamental difficulties of harmonic maps among Riemannian manifolds and of minimum immersions; comparable questions seem in physics, for instance within the classical concept of a-models. Non linear elasticity is one other instance in continuum mechanics, whereas Oseen-Frank idea of liquid crystals and Ginzburg-Landau idea of superconductivity require to regard variational difficulties with the intention to version fairly advanced phenomena. in most cases one is drawn to discovering strength minimizing representatives in homology or homotopy periods of maps, minimizers with prescribed topological singularities, topological fees, strong deformations i. e. minimizers in periods of diffeomorphisms or extremal fields. within the final or 3 a long time there was becoming curiosity, wisdom, and figuring out of the overall conception for this type of difficulties, sometimes called geometric variational difficulties. end result of the loss of a regularity concept within the non scalar case, not like the scalar one - or in different phrases to the prevalence of singularities in vector valued minimizers, frequently comparable with focus phenomena for the power density - and due to the actual relevance of these singularities for the matter being thought of the query of singling out a susceptible formula, or thoroughly knowing the importance of assorted susceptible formulations becames non trivial.
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