Comparison principles for viscosity solutions of elliptic branches of fully nonlinear equations independent of the gradient

Marco Cirant; Kevin R. Payne; Marco Cirant; Kevin R. Payne

doi:10.3934/mine.2021030

Mathematics in Engineering

2021, Volume 3, Issue 4: 1-45. doi: 10.3934/mine.2021030

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Comparison principles for viscosity solutions of elliptic branches of fully nonlinear equations independent of the gradient

Marco Cirant ¹,
Kevin R. Payne ^{2
,
,}

1.
Dipartimento di Matematica “T. Levi-Civita”, Università di Padova, Via Trieste 63, 35121–Padova, Italy
2.
Dipartimento di Matematica “F. Enriques”, Università di Milano, Via C. Saldini 50, 20133–Milano, Italy

Received: 25 February 2020 Accepted: 29 July 2020 Published: 03 August 2020

The validity of the comparison principle in variable coefficient fully nonlinear gradient free potential theory is examined and then used to prove the comparison principle for fully nonlinear partial differential equations which determine a suitable potential theory. The approach combines the notions of proper elliptic branches inspired by Krylov ^[18] with the monotonicity-duality method initiated by Harvey and Lawson ^[12]. In the variable coefficient nonlinear potential theory, a special role is played by the Hausdorff continuity of the proper elliptic map Θ which defines the potential theory. In the applications to nonlinear equations defined by an operator F, structural conditions on F will be determined for which there is a correspondence principle between Θ-subharmonics/superharmonics and admissible viscosity sub and supersolutions of the nonlinear equation and for which comparison for the equation follows from the associated compatible potential theory. General results and explicit models of interest in differential geometry will be examined. Examples of improvements with respect to existing results on comparison principles will be given.
- comparison principles,
- admissible viscosity solutions,
- proper elliptic branches,
- monotonicity,
- duality,
- special Lagrangian equation
Citation: Marco Cirant, Kevin R. Payne. Comparison principles for viscosity solutions of elliptic branches of fully nonlinear equations independent of the gradient[J]. Mathematics in Engineering, 2021, 3(4): 1-45. doi: 10.3934/mine.2021030

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Abstract

The validity of the comparison principle in variable coefficient fully nonlinear gradient free potential theory is examined and then used to prove the comparison principle for fully nonlinear partial differential equations which determine a suitable potential theory. The approach combines the notions of proper elliptic branches inspired by Krylov ^[18] with the monotonicity-duality method initiated by Harvey and Lawson ^[12]. In the variable coefficient nonlinear potential theory, a special role is played by the Hausdorff continuity of the proper elliptic map Θ which defines the potential theory. In the applications to nonlinear equations defined by an operator F, structural conditions on F will be determined for which there is a correspondence principle between Θ-subharmonics/superharmonics and admissible viscosity sub and supersolutions of the nonlinear equation and for which comparison for the equation follows from the associated compatible potential theory. General results and explicit models of interest in differential geometry will be examined. Examples of improvements with respect to existing results on comparison principles will be given.

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