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

```@article{Angenent1990ParabolicEF,
title={Parabolic equations for curves on surfaces Part I. Curves with \\$p\\$-integrable curvature},
author={Sigurd B. Angenent},
journal={Annals of Mathematics},
year={1990},
volume={132},
pages={451-483}
}```
• 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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The curve shortening problem, by now widely known, is to understand the evolution of regular closed curves γ: R/Z -> M moving according to the curvature normal vector: dy/dt = kN = -"the ZΛgradient
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