Singularity Behavior in Some Geometric Variational Problems

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This project lies in the area of geometric variational calculus, treating the behavior of singularities and energy concentration for various optimal or stationary functions, fields, measures, or geometric structures, possibly subject to constraints. The first specific class of projects involves continuing work with T. Riviere, on relations between the p energy of a map between Riemannian manifolds and its homotopy class. In case of nontorsion homotopy, our previous results on the geometric and topological structure of bubbles allows attack on open questions about the weak smooth approximability of Sobolev maps as well as refined questions about bubbling in p stationary maps and heat flows. We also are attacking higher order Sobolev spaces which seem more natural for certain homotopy classes, but for which basic approximation results and constructions have not been previously studied, A second class of projects involves continuing work with Thierry De Pauw on extending notions from geometric measure theory and solutions of Plateau-type problems in the context of chains in a metric space with coefficients in a general group. We consider a variety of mass-type functionals and the notion of a scan which generalizes the finite mass metric-space currents of Ambrosio-Kirchheim and the rectifiable and flat Euclidean-space G-chains of White. In the Euclidean space context we approximate the size functional of Almgren and prove optimal regularity for the minimizers of such approximate functionals. Solutions to many variational problems in both pure and applied mathematics often are forced to have singularities, that is, to involve regions where large oscillations occur. For example a nematic liquid crystal material in a spherical container whose optical axis is forced to point outward on the container necessarily will have singularities inside (observable through cross-polarizers or x-ray diffraction). In this example the optical axis has an energy density, which measures its local rate of change and whose integral tends to have a minimum value among all possible configurations. Our research proposes to understand the relationship between energies in such variational problems and the topological barriers imposed by the physics of these problems. We have derived new notions which allow the treatment and precise description of a wide variety of problems from soap films and their higher dimensional generalizations to optimal transport paths in various complex media. Geometric constraints which occur naturally in many physical problems have led to new mathematical and computational issues. In particular, two that we are studying are the constant cross-sectional area constraint exhibited in plant structure and gradient constraints in the microstructure formation in certain crystalline materials.