Parabolic equations for curves on surfaces Part I. Curves with $p$-integrable curvature

  title={Parabolic equations for curves on surfaces Part I. Curves with \$p\$-integrable curvature},
  author={Sigurd B. Angenent},
  journal={Annals of Mathematics},
  • S. Angenent
  • Published 1 November 1990
  • Mathematics
  • Annals of Mathematics
This is the first of a two-part paper in which we develop a theory of parabolic equations for curves on surfaces which can be applied to the so-called curve shortening of flow-by-mean-curvature problem, as well as to a number of models for phase transitions in two dimensions. We introduce a class of equations for which the initial value problem is solvable for initial data with p-integrable curvature, and we also give estimates for the rate at which the p-norms of the curvature must blow up, if… 
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