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Allard-type boundary regularity for C^(1,α) boundaries

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arxiv 1008.4728 v2 pith:KDHXBPF4 submitted 2010-08-27 math.AP

Allard-type boundary regularity for C^(1,α) boundaries

classification math.AP
keywords boundaryallardalphamonotonicityregularityballsfirstformulae
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In this paper we show boundary monotonicity formulae for rectifiable varifolds having a $C^{1,\alpha}$ "boundary". In particular, we show that the area ratios of balls centered at this "boundary'' satisfy a nice monotonicity formula, similar to that for interior balls (proved in Allard's paper "On the first variation of a varifold''). This extends the boundary monotonicity formulae of Allard (see "On the first variation of a varifold- boundary behavior''), which require that the boundary is $C^{1,1}$. As a corollary, Allard's boundary regularity results extend to this case and provide a regularity result for rectifiable varifolds with a $C^{1,\alpha}$ ``boundary''.

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