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Hessian Estimates for Nondivergence Parabolic and Elliptic Equations with Partially BMO Coefficients

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Abstract

In this paper, we mainly prove a local Calderón–Zygmund estimate in the Lorentz spaces with a variable power \(p(\cdot )\) to the Hessian of nondivergence parabolic equations \(u_{t}(x,t)-a_{ij}(x,t)D_{ij}u(x,t)=f(x,t)\), under assumptions that the variable exponent \(p(\cdot )\) is \(\log \)-Hölder continuous, and the coefficient is a small partially BMO matrix which means that \(a_{ij}(x,t)\) is merely measurable in one of spatial variables and have small BMO semi-norms with respect to other variables. In addition, we also derive a similar result for nondivergence elliptic equations \(a_{ij}(x)D_{ij}u(x)=f(x)\) with small partially BMO coefficients.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions, which improved the quality of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China

    Junjie Zhang & Shenzhou Zheng

  2. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050016, Hebei, China

    Junjie Zhang

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  1. Junjie Zhang
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  2. Shenzhou Zheng
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Correspondence to Shenzhou Zheng.

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This paper was supported by Hebei Normal University of Science and Technology Grant No. L2019B02 and by Natural Science Foundation of Hebei Province Grant No. A2019205218.

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Zhang, J., Zheng, S. Hessian Estimates for Nondivergence Parabolic and Elliptic Equations with Partially BMO Coefficients. Results Math 75, 21 (2020). https://doi.org/10.1007/s00025-019-1147-z

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