Abstract:
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 particular, 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.
We can apply this result to the minimization of anisotropic energies among
families of d-rectifiable closed subsets of $\mathbb{R}^n$. 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 prove an anisotropic
counterpart of Allard's compactness result for integral varifolds.
To conclude, we give some ideas of an ongoing project, which relies on the
presented rectifiability theorem.
The main result is a joint work with G. De Philippis and F. Ghiraldin. |