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Global Gradient Estimates of Very Weak Solutions for a General Class of Quasilinear Elliptic Equations

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Abstract

We are concerned with global gradient estimates of very weak solutions for a general class of quasilinear elliptic equations with nonstandard growth. We establish such estimates under basic structure assumptions on the nonlinearity and natural geometric assumption on the complement of the bounded domain. Our approach is based on a generalized Lipschitz truncation alongside a generalized self-improving property of thick sets. This work provides a careful analysis of the thickness described in Adimurthi and Phuc (Calc Var Partial Differ Equ 54:3107–3139, 2015), Harjulehto and Hästö (Z Anal Anwend 38(1):73–96, 2019), Heinonen et al (Potential theory of degenerate elliptic equations, Oxford University Press, Oxford, 1993).

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Acknowledgements

We thank an anonymous referee for the careful reading of the earlier version and providing helpful comments and suggestions which improved the manuscript. S. Byun was supported by the National Research Foundation of Korea Grant (NRF-2021R1A4A1027378). M. Lim was supported by the National Research Foundation of Korea Grant (NRF-2022R1A2C1009312).

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  1. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 151-747, Korea

    Sun-Sig Byun

  2. Research Institute of Mathematics, Seoul National University, Seoul, 151-747, Korea

    Minkyu Lim

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  1. Sun-Sig Byun
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Byun, SS., Lim, M. Global Gradient Estimates of Very Weak Solutions for a General Class of Quasilinear Elliptic Equations. J Geom Anal 33, 156 (2023). https://doi.org/10.1007/s12220-023-01210-3

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