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Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition. (English) Zbl 0851.35053

Summary: We study the behaviour of compact, smooth, immersed manifolds with boundary which move under the mean curvature flow in Euclidean space. We thereby prescribe the Neumann boundary condition in a purely geometric manner by requiring a vertical contact angle between the unit normal fields of the immersions and a given, smooth hypersurface \(\Sigma\). We deduce a very sharp local gradient bound depending only on the curvature of the immersions and \(\Sigma\). Combining this with a short time existence result, we obtain the existence of a unique solution to any given smooth initial and boundary data. This solution either exists for any \(t> 0\) or on a maximal finite time interval \([0, T]\) such that the curvature explodes as \(t\to T\).

MSC:

35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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