Boundedness and higher integrability of minimizers to a class of two-phase free boundary problems under non-standard growth conditions

Jiayin Liu; Jun Zheng; Jiayin Liu; Jun Zheng

doi:10.3934/math.2024904

AIMS Mathematics

2024, Volume 9, Issue 7: 18574-18588. doi: 10.3934/math.2024904

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Research article

Boundedness and higher integrability of minimizers to a class of two-phase free boundary problems under non-standard growth conditions

Jiayin Liu ^1,2,
Jun Zheng ^{3,4
,
,}

1.
School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China
2.
School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China
3.
School of Mathematics, Southwest Jiaotong University, Chengdu 611756, China
4.
Department of Electrical Engineering, Polytechnique Montréal, P. O. Box 6079, Station Centre-Ville, Montreal, QC, Canada

Received: 17 March 2024 Revised: 28 May 2024 Accepted: 29 May 2024 Published: 03 June 2024
MSC : 35A15, 35J60

In this paper, we are concerned with the existence, boundedness, and integrability of minimizers of heterogeneous, two-phase free boundary problems $ \mathcal {J}_{\gamma}(u) = \int_{\Omega}\left(f(x, \nabla u)+\lambda_{+}(u^{+})^{\gamma}+\lambda_{-}(u^{-})^{\gamma}+gu\right)\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 $, respectively.
- free boundary problem,
- two-phase,
- integrability,
- non-standard growth,
- minimizer
Citation: Jiayin Liu, Jun Zheng. Boundedness and higher integrability of minimizers to a class of two-phase free boundary problems under non-standard growth conditions[J]. AIMS Mathematics, 2024, 9(7): 18574-18588. doi: 10.3934/math.2024904

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Abstract

In this paper, we are concerned with the existence, boundedness, and integrability of minimizers of heterogeneous, two-phase free boundary problems $ \mathcal {J}_{\gamma}(u) = \int_{\Omega}\left(f(x, \nabla u)+\lambda_{+}(u^{+})^{\gamma}+\lambda_{-}(u^{-})^{\gamma}+gu\right)\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 $, respectively.

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