Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

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

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

This contribution is part of the Special Issue: Partial Differential Equations from theory to applications—Dedicated to Alberto Farina, on the occasion of his 50th birthday
Guest Editors: Serena Dipierro; Luca Lombardini
Link: www.aimspress.com/mine/article/5752/special-articles

Special Issues: Partial Differential Equations from theory to applications—Dedicated to Alberto Farina, on the occasion of his 50th birthday

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.
  Article Metrics


1. Aubin JP, Cellina A (1984) Differential Inclusions: Set Valued Maps and Viability Theory, Berlin: Springer-Verlag.

2. Bardi M, Mannucci P (2006) On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Commun Pure Appl Anal 73: 709-731.

3. Barles G, Busca J (2001) Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Commun Part Diff Eq 26: 2323-2337.    

4. Birindelli I, Payne KR (2021) Principal eigenvalues for k-Hessian operators by maximum principle methods. Mathematics in Engineering 3: 1-37.

5. Burago D, Burago Y, Ivanov S (2001) A Course in Metric Geometry, Providence, RI: American Mathematical Society.

6. Cheng SY, Yau ST (1986) Complete affine hypersurfaces. Part I. The completeness of affine metrics. Commun Pure Appl Math 39: 839-866.

7. Cirant M, Harvey FR, Lawson HB, et al. (2020) Comparison principles by monotonicity and duality for constant coefficient nonlinear potential theory and PDEs. Preprint.

8. Cirant M, Payne KR (2017) On viscosity solutions to the Dirichlet problem for elliptic branches of nonhomogeneous fully nonlinear equation. Publ Mat 61: 529-575.    

9. Crandall MG, Ishii H, Lions PL (1992) User's guide to viscosity solutions of second order partial differential equations. B Am Math Soc 27: 1-67.    

10. Dinew S, Do HS, Tô TD (2019) A viscosity approach to the Dirichlet problem for degenerate complex Hessian-type equations. Anal PDE 2: 505-535.

11. Harvey FR, Lawson Jr HB (1982) Calibrated geometries. Acta Math 148: 47-157.    

12. Harvey FR, Lawson Jr HB (2009) Dirichlet duality and the nonlinear Dirichlet problem. Commun Pure Appl Math 62: 396-443.    

13. Harvey FR, Lawson Jr HB (2011) Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds. J Differ Geom 88: 395-482.    

14. Harvey FR, Lawson Jr HB (2016) Notes on the differentiation of quasi-convex functions. arXiv:1309.1772v3.

15. Harvey FR, Lawson Jr HB (2018) The inhomogeneous Dirichlet Problem for natural operators on manifolds. arXiv:1805.111213v1.

16. Harvey FR, Lawson Jr HB (2019) Pseudoconvexity for the special Lagrangian potential equation. Preprint.

17. Kawohl B, Kutev N (2007) Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations. Commun Part Diff Eq 32: 1209-1224.    

18. Krylov NV (1995) On the general notion of fully nonlinear second-order elliptic equations. T Am Math Soc 347: 857-895.    

19. Slodkowski Z (1984) The Bremermann-Dirichlet problem for q-plurisubharmonic functions. Ann Scuola Norm Sup Pisa Cl Sci 11: 303-326.

© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved