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Article Dans Une Revue Communications on Pure and Applied Mathematics Année : 2019

An Epiperimetric Inequality for the Regularity of Some Free Boundary Problems: The 2‐Dimensional Case

Résumé

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 $C^{1,\alpha}$ 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 each connected component of $\{|u|=0\}$ might have cusps, but they must be a finite number.

Dates et versions

hal-02013993 , version 1 (11-02-2019)

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Luca Spolaor, Bozhidar Velichkov. An Epiperimetric Inequality for the Regularity of Some Free Boundary Problems: The 2‐Dimensional Case. Communications on Pure and Applied Mathematics, 2019, 72 (2), pp.375-421. ⟨10.1002/cpa.21785⟩. ⟨hal-02013993⟩
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