An epiperimetric inequality for the regularity of some free boundary problems: the $2$-dimensional case

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.