$\displaystyle \qquad\qquad\qquad
J_{TP}(u,D):=\int_{D} \vert\nabla u\vert^2\,dx+\lambda^2_+\vert\Omega_u^+\cap D\vert+\lambda^2_-\vert\Omega_u^-\cap D\vert,$

where $\lambda_+>0$ and $\lambda_->0$ are fixed constants, $\Omega_u^+=\{u>0\}$ and $\Omega_u^-=\{u<0\}$.

In this talk, we will present some recent results on the regularity of the free boundary $\partial\Omega_u^+\cup \partial\Omega_u^-\cap D$ of
local minimizers $u$ of the two-phase functional $J_{TP}$ in a fixed open domain $D\subset\mathbb{R}^d$. Precisely, we will show that:

$-$ In dimension $d=2$, the free boundaries $\partial\Omega_u^+\cap D$ and $\partial\Omega_u^-\cap D$ are $C^{1,\alpha}$-regular curves (this result was proved in recent works of Spolaor-V. and Spolaor-Trey-V.);

$-$ In dimension $d>2$, the free boundaries $\partial\Omega_u^+\cap D$ and $\partial\Omega_u^-\cap D$ are $C^{1,\alpha}$-regular, up to a (possibly empty) one-phase singular set of lower dimension (De Philippis-Spolaor-V.).

In particular, these results complete (in any dimension) the analysis of the two-phase free boundaries started by Alt, Caffarelli and Friedman in [TAMS, 1984]
http://cvgmt.sns.it/seminar/712/