Sala Seminari (Dip. Matematica).
We present our recent extension of Allard’s celebrated rectifiability theorem to the setting of varifolds with locally bounded first variation with respect to an anisotropic integrand. In partic- ular, we identify a necessary and sufficient condition on the integrand to obtain the rectifiability of every d-dimensional varifold with locally bounded first variation and positive d-dimensional density. In codimension one, this condition is shown to be equivalent to the strict convexity of the integrand with respect to the tangent plane. We can apply this result to the minimization of anisotropic energies among families of d- rectifiable closed subsets of Rn, closed under Lipschitz deformations (in any dimension and codimension). Easy corollaries of this compactness result are the solutions to three formulations of the Plateau problem: one introduced by Reifenberg, one proposed by Harrison and Pugh and another one studied by Guy David. Moreover, we apply the rectifiability theorem to the energy minimization in classes of vari- folds and to a compactness result of integral varifolds in the anisotropic setting. Finally, we show some connections of the Plateau problem with branched transport, min- imizing concave costs among 1-dimensional currents. In particular, we prove a stability result for the optimal transports. References \t G. De Philippis, A. De Rosa, and F. Ghiraldin. Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies. 2016. Accepted on Comm. Pure Appl. Math. \t \t G. De Philippis, A. De Rosa, and F. Ghiraldin. A direct approach to Plateau’s problem in any codimension. Adv. in Math., 288:59–80, January 2015. \t \t C. De Lellis, A. De Rosa, and F. Ghiraldin. A direct approach to the anisotropic Plateau’s problem. 2016. Available at http://arxiv.org/abs/1602.08757 \t \t A. De Rosa. Minimization of anisotropic energies in classes of rectifiable varifolds. 2016. Available at http://arxiv.org/abs/1611.07929 \t \t G. De Philippis, A. De Rosa, and F. Ghiraldin. Existence results for minimizers of para- metric elliptic functionals. 2016. In preparation. \t \t M. Colombo, A. De Rosa, A. Marchese, and S. Stuvard. On the lower semicon- tinuous envelope of functionals defined on polyhedral chains. 2017. Available on: https://arxiv.org/abs/1703.01938. \t \t M. Colombo, A. De Rosa, and A. Marchese. Improved stability of optimal traffic paths. Available on arxiv at https://arxiv.org/abs/1701.07300, 2017. \t