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

## Abstracts

 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
From: Jun Zheng
DOI：10.12074/201809.00180
Recommended references： Jun Zheng.(2018).Regularity in the two-phase free boundary problems under non-standard growth conditions.[ChinaXiv:201809.00180] (Click&Copy)
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