Abstract
We derive global gradient estimates for \(W^{1,p}_0(\Omega )\)-weak solutions to quasilinear elliptic equations of the form
over n-dimensional Reifenberg flat domains. The nonlinear term of the elliptic differential operator is supposed to be small-BMO with respect to x and merely continuous in u. Our result highly improves the known regularity results available in the literature. Actually, we are able not only to weaken the Lipschitz continuity with respect to u of the nonlinearity to only uniform continuity, but we also find a very lower level of geometric assumption on the boundary of the domain to ensure a global character of the gradient estimates obtained.
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Acknowledgements
S.-S. Byun was supported by NRF-2017R1A2B2003877. D.K. Palagachev is member of INdAM/GNAMPA. P. Shin was supported by NRF-2017R1D1A1B03036340. The authors thank the referee for the careful reading of the first version of this manuscript and for the valuable comments and suggestions offered.
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Communicated by L. Ambrosio.
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Byun, SS., Palagachev, D.K. & Shin, P. Global Sobolev regularity for general elliptic equations of p-Laplacian type. Calc. Var. 57, 135 (2018). https://doi.org/10.1007/s00526-018-1408-9
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DOI: https://doi.org/10.1007/s00526-018-1408-9