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Mean curvature flow with obstacles. (English) Zbl 1252.49072
Summary: We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity of the obstacles, in the two-dimensional case, we show existence and uniqueness of a regular solution before the onset of singularities. Finally, we discuss an application of this result to the positive mean curvature flow.

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
35R37 Moving boundary problems for PDEs
35R45 Partial differential inequalities and systems of partial differential inequalities
49J40 Variational inequalities
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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