Anisotropic counterpart of Allard’s rectifiability theorem and applications.
Anisotropic counterpart of Allard’s rectifiability theorem and applications.
We present our recent extension of Allard's celebratedÌýrectifiability theoremÌý to the setting ofÌý varifoldsÌý with locally boundedÌýfirst variation with respect to an anisotropic integrand. In particular,Ìýwe identify a necessary and sufficient condition on the integrand toÌýobtain the rectifiability of every d-dimensional varifold with locallyÌýbounded first variation and positive d-dimensional density.
We can apply this result to the minimization of anisotropic energies amongÌýfamilies of d-rectifiable closed subsets of $\mathbb{R}^n$. EasyÌýcorollaries of this compactness result are the solutions to threeÌýformulations of the Plateau problem: one introduced by Reifenberg, oneÌýproposed by Harrison and PughÌýand another one studied by Guy David.
Moreover, we apply the rectifiability theorem to prove an anisotropicÌýcounterpart of Allard's compactness result for integral varifolds.
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