Regularity in the two-phase free boundary problems under non-standard growth conditions

Abstract

In this paper, we prove several regularity results for the heterogeneous, two-phase free boundary problems ${\mathcal{J}}_{\gamma }(u)={\int }_{\Omega }(f(x,\nabla u)+{\lambda }_{+}(u^{+}{)}^{\gamma }+{\lambda }_{-}(u^{-}{)}^{\gamma }+gu)\text{d}x\to \text{min}$ 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 $\gamma =1$. Our results hold not only in the degenerate case of $p>2$ for $p-$Laplace equations, but also in the singular case of $1<p<2$, which are extensions of \cite{LdT}.

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