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

International audienceUsing 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