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

Abstract

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 $$J(v,\hat{I}\text{c◯})={\int }_{\hat{I}}\text{c◯}\left(|D{v}^{+}{|}^p+|D{v}^{-}{|}^q\right)dx.$$ 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 $$(q-1)|D{u}^{-}{|}^q=(p-1)|D{u}^{+}{|}^p,on\partial \{u>0\}.$$ We show that the free boundary is $C^{1,\hat{I}\pm }$ a.e. with respect to the measure $\hat{I}{”}_p{u}^{+}$, whose support is of $\ddot{I}ƒ$-finite $(n-1)$-dimensional Hausdorff measure.

35 pages