  In this article, we study a class of fully nonlinear double-divergence
systems with free boundaries associated with a minimization problem. The
variational structure of Hessian-dependent functional plays a fundamental role
in proving the existence of the minimizers and then the existence of the
solutions for the system. In addition, we establish gains of the integrability
for the double-divergence equation. Consequently, we improve the regularity for
the fully nonlinear equation in Sobolev and H\"older spaces.
