Triple Junctions in Multi-Phase Obstacle type Problems

We study the regularity of the free boundary for a multi-phase obstacle-type problem associated to quadrature domains, which arises from the minimization of the energy $$J(U,F):={\int }_{B_1}(\frac{1}{2}\mathrm{\mid }\mathrm{\nabla }U{\mathrm{\mid }}^2+F\cdot U)dx$$ among all competitors $U=(u^1,\dots ,u^N)$ with non-negative components $u^i\ge 0$ with disjoint supports, and constant forcing terms $f^i>0.$ We focus on the geometry of the free boundary at points where at least three distinct components meet (triple junctions).