## 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

### Keywords:

mean curvature flow; Neumann boundary condition
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### References:

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