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Interior gradient estimates for quasilinear elliptic equations

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Abstract

We study quasilinear elliptic equations of the form \(\text{ div }\,\mathbf {A}(x,u,\nabla u) = \text{ div }\,\mathbf {F}\) in bounded domains in \({\mathbb R}^n\), \(n\ge 1\). The vector field \(\mathbf {A}\) is allowed to be discontinuous in x, Lipschitz continuous in u and its growth in the gradient variable is like the p-Laplace operator with \(1<p<\infty \). We establish interior \(W^{1,q}\)-estimates for locally bounded weak solutions to the equations for every \(q>p\), and we show that similar results also hold true in the setting of Orlicz spaces. Our regularity estimates extend results which are only known for the case \(\mathbf {A}\) is independent of u and they complement the well-known interior \(C^{1,\alpha }\)- estimates obtained by DiBenedetto (Nonlinear Anal 7(8):827–850, 1983) and Tolksdorf (J Differ Equ 51(1):126–150, 1984) for general quasilinear elliptic equations.

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Acknowledgments

The authors would like to thank Luan Hoang for fruitful discussions. T. Nguyen and T. Phan gratefully acknowledge the supports of the Grants # 318995 and # 354889, respectively, from the Simons Foundation.

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Authors and Affiliations

  1. Department of Mathematics, University of Akron, 302 Buchtel Common, Akron, OH, 44325-4002, USA

    Truyen Nguyen

  2. Department of Mathematics, University of Tennessee, 227 Ayress Hall, 1403 Circle Drive, Knoxville, TN, 37996, USA

    Tuoc Phan

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  1. Truyen Nguyen
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  2. Tuoc Phan
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Correspondence to Truyen Nguyen.

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Communicated by L. Ambrosio.

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Nguyen, T., Phan, T. Interior gradient estimates for quasilinear elliptic equations. Calc. Var. 55, 59 (2016). https://doi.org/10.1007/s00526-016-0996-5

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