Zeitschrift für Analysis und ihre Anwendungen


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Volume 36, Issue 1, 2017, pp. 37–47
DOI: 10.4171/ZAA/1578

Published online: 2017-01-06

Regularity of Minimizers in the Two-Phase Free Boundary Problems in Orlicz–Sobolev Spaces

Jun Zheng[1], Binhua Feng[2] and Peihao Zhao[3]

(1) Southwest Jiaotong University, Chengdu, China
(2) Northwest Normal University, Lanzhou, China
(3) Lanzhou University, China

In this paper, we consider the optimization problem of minimizing $\mathcal {J}(u)=\int_{\Omega}(G(|\nabla u|)+\lambda_{+}(u^{+})^{\gamma}+\lambda_{-}(u^{-})^{\gamma}+fu)\text{d}x$ in the class of functions $W^{1,G}(\Omega)$ with $u - \varphi \in W^{1,G}_{0}(\Omega)$ for a given function $\varphi$, where $W^{1,G}(\Omega)$ is the class of weakly differentiable functions with $\int_{\Omega} G(|\nabla u|)\text{d}x<\infty$. The conditions on the function $G$ allow for a different behavior at $0$ and at $\infty$. For $0<\gamma \leq 1$, we prove that every minimizer $u$ of $\mathcal {J}(u)$ is $C^{1,\alpha}_{loc}$-continuous.

Keywords: Free boundary problem, regularity, minimizer, Orlicz spaces

Zheng Jun, Feng Binhua, Zhao Peihao: Regularity of Minimizers in the Two-Phase Free Boundary Problems in Orlicz–Sobolev Spaces. Z. Anal. Anwend. 36 (2017), 37-47. doi: 10.4171/ZAA/1578