Boundedness and higher integrability of minimizers to a class of two-phase free boundary problems under non-standard growth conditions

In this paper, we are concerned with the existence, boundedness, and integrability of minimizers of heterogeneous, two-phase free boundary problems ${\mathcal{J}}_{\gamma }(u)={\int }_{\mathrm{\Omega }}\left(f(x,\mathrm{\nabla }u)+{\lambda }_{+}(u^{+}{)}^{\gamma }+{\lambda }_{-}(u^{-}{)}^{\gamma }+gu\right)\text{d}x\to \text{min}$ under non-standard growth conditions. Included in such problems are heterogeneous jets and cavities of Prandtl-Batchelor type with $\gamma =0$, chemical reaction problems with $0<\gamma <1$, and obstacle type problems with $\gamma =1$, respectively.