2015/06/04 by Jenny Harrison, Harrison, J., Harrison Pugh +1
Mathematics · #49Q10 #49Q20 #53C42 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1506.01692
openalex publication_date 2015/06/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We prove existence and regularity of minimizers for Hölder densities over general surfaces of arbitrary dimension and codimension in \(\Rn \), satisfying a cohomological boundary condition, providing a natural dual to Reifenberg's Plateau problem. We generalize and extend methods of Reifenberg, Besicovitch, and Adams, in particular we generalize a particular type of minimizing sequence used by Reifenberg (whose limits have nice properties, including lower bounds on lower density and finite Hausdorff measure,) prove such minimizing sequences exist, and develop cohomological spanning conditions. Our cohomology lemmas are dual versions of the homology lemmas in the celebrated appendix by Adams found in Reifenberg's 1960 paper.