An Epiperimetric Inequality for the Regularity of Some Free Boundary Problems: The 2-Dimensional Case |
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| Authors: | Luca Spolaor Bozhidar Velichkov |
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| Institution: | 1. Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA;2. Laboratoire Jean Kuntzmann, Université Grenoble Alpes, Bâtiment IMAG, 700 Avenue Centrale, 38401 Saint-Martin-d'Hères, France |
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| Abstract: | Using a direct approach, we prove a two-dimensional epiperimetric inequality for the one-phase problem in the scalar and vectorial cases and for the double-phase problem. From this we deduce, in dimension 2, the C1,α regularity of the free boundary in the scalar one-phase and double-phase problems, and of the reduced free boundary in the vectorial case, without any restriction on the sign of the component functions. Furthermore, we show that in the vectorial case each connected component of  might have cusps, but they must be a finite number. © 2018 Wiley Periodicals, Inc. |
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