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Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations

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Abstract

We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison principle, which we refer to as the principle of propagation of touching points, for operators of the form \(\nabla ^2 \psi + L(x,\psi ,\nabla \psi )\) which are non-decreasing in \(\psi \).

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Acknowledgements

Li is partially supported by NSF Grant DMS-1501004. Wang is partially supported by NNSF (11701027) and Beijing Institute of Technology Research Fund Program for Young Scholars.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    YanYan Li

  2. Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, NJ, 08854, USA

    YanYan Li

  3. Mathematical Institute and St Edmund Hall, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    Luc Nguyen

  4. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Bo Wang

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  1. YanYan Li
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  2. Luc Nguyen
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  3. Bo Wang
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Correspondence to Bo Wang.

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Communicated by L. Ambrosio.

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Li, Y., Nguyen, L. & Wang, B. Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations. Calc. Var. 57, 96 (2018). https://doi.org/10.1007/s00526-018-1369-z

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