Published Paper
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Inserted: 8 feb 2017
Last Updated: 13 nov 2026

Journal: Comm. Pure Appl. Math.
Pages: 32
Year: 2019
Doi: https://doi.org/10.1002/cpa.21785

Abstract:

Using a direct approach, we prove a $2$-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 $C^{1,\alpha}$ 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 the free boundary can end in a cusp.

Keywords: Regularity of free boundary

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