  In this paper we study nonnegative minimizers of general degenerate elliptic
functionals, $\int F(X,u,Du) dX \to \min$, for variational kernels $F$ that are
discontinuous in $u$ with discontinuity of order $\sim \chi_{\{u > 0 \}}$. The
Euler-Lagrange equation is therefore governed by a non-homogeneous, degenerate
elliptic equation with free boundary between the positive and the zero phases
of the minimizer. We show optimal gradient estimate and nondegeneracy of
minima. We also address weak and strong regularity properties of free boundary.
We show the set $\{u > 0 \}$ has locally finite perimeter and that the reduced
free boundary, $\partial_\text{red} \{u > 0 \}$, has $\mathcal{H}^{n-1}$-total
measure. For more specific problems that arise in jet flows, we show the
reduced free boundary is locally the graph of a $C^{1,\gamma}$ function.
