A transmission problem for $(p,q)$-Laplacian

In this paper, we consider a double-phase problem characterised by a transmission that takes place across the zero level "surface" of the minimiser of the functional $$J(v,\Omega) = \int_\Omega \left( |D v^+|^p + |D v^-|^q \right) dx.$$ We prove that a minimiser exists, and is H\"older continuous, whence using an intrinsic variation we prove a weak formulation of the free boundary condition across the zero level surface, formally represented by $$(q-1)|D u^-|^q = (p-1) |D u^+|^p, \quad \hbox{on } \partial \{u > 0\}.$$ We show that the free boundary is $C^{1,\alpha}$ a.e. with respect to the measure $\Delta_p u^+$, whose support is of $\sigma$-finite $(n-1)$-dimensional Hausdorff measure.