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Local \(C^{1,\beta }\)-regularity at the boundary of two dimensional sliding almost minimal sets in \(\mathbb{R}^3 \). (English) Zbl 1459.49015

The author proves a \(C^{1,\beta}\)-regularity result on the boundary for two-dimensional sliding almost minimal sets in \({\mathbb R}^3\). This effect may apply to the regularity of the soap films at the boundary. Guy David proposed to consider the Plateau problem with sliding boundary conditions, since it is very natural to the soap films. This result may also lead to the existence of a solution to the Plateau problem with sliding boundary conditions proposed in the case that the boundary is a 2-dimensional smooth submanifold.

MSC:

49K99 Optimality conditions
49Q20 Variational problems in a geometric measure-theoretic setting
49J99 Existence theories in calculus of variations and optimal control
49N60 Regularity of solutions in optimal control
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