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In this paper we prove C 1,α regularity (near flat points) of the free boundary ∂{u > 0} ∩ Ω in the Alt-Caffarelli type minimum problem for the p-Laplace operator: (2000): 35R35, 35J60 Mathematics Subject Classification

We give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren's monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary points, the regularity of the regular set and the structure of the singular set.2000 Mathematics Subject Classification. Primary 35R35, 35K85.

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