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Weighted \(W^{1,2}_{p(\cdot )}\)-Estimate for Fully Nonlinear Parabolic Equations with a Relaxed Convexity

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Abstract

We devote this paper to global estimate in weighted variable exponent Sobolev spaces for fully nonlinear parabolic equations under a relaxed convexity condition. It is assumed that the associated variable exponent is log-Hölder continuous, the weight belongs to certain Muckenhoupt class concerning the variable exponent, the leading part of nonlinearity satisfies a relaxed convexity in Hessian and is of VMO condition in space-time variables, and the boundary of underlying domain satisfies \(C^{1,1}\)-smooth. Our key strategy is to utilize a unified approach based on the generalized versions of Fefferman–Stein theorem of the sharp functions and extrapolation to establish the estimates of \(D^{2}u\) and \(D_{t} u\) within the framework of weighted variable exponent Lebesgue spaces.

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Funding

H. Tian was supported by National Natural Science Foundation of China Grant No. 11901429, and S. Zheng was supported by National Natural Science Foundation of China Grant No. 12071021.

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Authors and Affiliations

  1. College of Science, Tianjin University of Technology, Tianjin, 300384, China

    Hong Tian

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

    Shenzhou Zheng

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  1. Hong Tian
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Hong Tian and Shenzhou Zheng wrote the main manuscript text and reviewed the manuscript.

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Correspondence to Shenzhou Zheng.

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Tian, H., Zheng, S. Weighted \(W^{1,2}_{p(\cdot )}\)-Estimate for Fully Nonlinear Parabolic Equations with a Relaxed Convexity. Mediterr. J. Math. 21, 120 (2024). https://doi.org/10.1007/s00009-024-02659-4

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