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

Jun Zheng

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arXiv · 1809.08519

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

Jun Zheng

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Abstract

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 $1<p<2$, which are extensions of \cite{LdT}.

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Jun Zheng. 2018-09-23. Regularity in the two-phase free boundary problems under non-standard growth conditions. https://arxiv.org/abs/1809.08519

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