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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 $$ J(v,Ω) = \int_Ω\left( |D v^+|^p + |D v^-|^q \right) dx. $$ We prove that a minimiser exists, and is Hölder 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 $C^{1,α}$ a.e. with respect to the measure $Δ_p u^+$, whose support is of $σ$-finite $(n-1)$-dimensional Hausdorff measure.