摘要: In this paper, we prove several regularity results for the heterogeneous, two-phase free boundary problems $\mathcal {J}_{\gamma}(u)=\int_{\Omega}\big(f(x,\nabla
u)+(\lambda_{+}(u^{+})^{\gamma}+\lambda_{-}(u^{-})^{\gamma})+gu\big)\text{d}x\rightarrow \text{min}$
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 $\gamma=1$. Our results hold not only in
the degenerate case of $p> 2$ for $p-$Laplace equations, but
also in the singular
case of $1
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