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Global gradient estimates for non-uniformly elliptic equations

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Abstract

We consider a nonlinear and non-uniformly elliptic problem in divergence form on a bounded domain. The problem under consideration is characterized by the fact that its ellipticity rate and growth radically change with the position, which provides a model for describing a feature of strongly anisotropic materials. We establish the global Calderón–Zygmund type estimates for the distributional solution in the case that the boundary of the domain is of class \(C^{1,\beta }\) for some \(\beta >0\).

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Acknowledgements

The authors would like to thank the referee for the very valuable suggestions and comments which led to improvement of the paper. S. Byun was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (NRF-2015R1A2A1A15053024). J. Oh was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea Government (NRF-2015R1A4A1041675).

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Authors and Affiliations

  1. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 08826, Korea

    Sun-Sig Byun

  2. Department of Mathematical Sciences, Seoul National University, Seoul, 08826, Korea

    Jehan Oh

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  1. Sun-Sig Byun
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Correspondence to Jehan Oh.

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Communicated by N. Trudinger .

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Byun, SS., Oh, J. Global gradient estimates for non-uniformly elliptic equations. Calc. Var. 56, 46 (2017). https://doi.org/10.1007/s00526-017-1148-2

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