Abstract
We continue our investigation of the “level-set” technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature. The principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by K. Brakke: we prove that almost every level set of the solution to the mean curvature evolution PDE is in fact aunit-density varifold moving according to its mean curvature. In particular, a.e. level set is endowed with a kind of “geometric structure.” The proof utilizes compensated compactness methods to pass to limits in various geometric expressions.
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L.C.E. supported in part by NSF Grant DMS-89-03328. J.S. supported in part by NSF Grant DMS-85-01952 and DOE Grant DE-FG02-86ER25015.
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Evans, L.C., Spruck, J. Motion of level sets by mean curvature IV. J Geom Anal 5, 77–114 (1995). https://doi.org/10.1007/BF02926443
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DOI: https://doi.org/10.1007/BF02926443