# 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} }

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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