Skip to main content
SpringerLink
Log in

On the comparison principle for second order elliptic equations without first and zeroth order terms

  • Published:
Nonlinear Differential Equations and Applications NoDEA Aims and scope Submit manuscript

Abstract

We consider the comparison principle for semicontinuous viscosity sub- and supersolutions of second order elliptic equations on the form \(F({\mathcal {H}}w,x) = 0\). A structural condition on the operator is presented that seems to unify the different existing theories. A new result is obtained and the proofs of the classical results are simplified.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. Unless \(x\mapsto {{\,\textrm{dist}\,}}(X_0,\Theta _{+}(x))\) is continuous at \(x_0\). Then a counterexample can be produced in the same way as in the proof (i) \(\Rightarrow \) (ii) of Proposition 2.6.

  2. At least not in our setting with a classical definition of viscosity sub- and supersolutions. There is, however, a theory for \(L^p\)-viscosity solutions where this can be allowed since the ingredients then are interpreted only outside sets of measure zero. See e.g. [6].

References

  1. Beer, G.: Topologies on closed and closed convex sets. Mathematics and its Applications, vol. 268. Kluwer Academic Publishers Group, Dordrecht (1993)

  2. Bhatia, R.: Matrix analysis. Graduate Texts in Mathematics, vol. 169. Springer, New York (1997)

  3. Bardi, M., Mannucci, P.: On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Commun. Pure Appl. Anal. 5(4), 709–731 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brustad, K.K.: Segre’s theorem. An analytic proof of a result in differential geometry. Asian J. Math 25(3), 321–340 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brustad,Karl K.: Counterexamples to the comparison principle in the special Lagrangian potential equation. arXiv:2206.09373 (2022)

  6. Caffarelli, L., Crandall, M.G., Kocan, M., Swiech, A.: On viscosity solutions of fully nonlinear equations with measurable ingredients. Commun. Pure Appl. Math. 49(4), 365–397 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cirant, M., Harvey, F.R, Lawson, H.B., Payne, K.R.: Comparison principles by monotonicity and duality for constant coefficient nonlinear potential theory and pdes. arXiv:2009.01611v1 (2020)

  8. Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cirant, M., Payne, K.R.: On viscosity solutions to the Dirichlet problem for elliptic branches of inhomogeneous fully nonlinear equations. Publ. Mat. 61(2), 529–575 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cirant, M., Payne, K.R.: Comparison principles for viscosity solutions of elliptic branches of fully nonlinear equations independent of the gradient. Math. Eng. 3(4), 45 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Crandall, M.G.: Viscosity solutions: a primer. In: Viscosity solutions and applications (Montecatini Terme, 1995), volume 1660 of Lecture Notes in Math., pp. 1–43. Springer, Berlin (1997)

  12. Harvey, R., Lawson, H.B.: Jr. Calibrated geometries. Acta Math. 148, 47–157 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. Reese Harvey, F., Blaine Lawson, H.: Dirichlet duality and the nonlinear Dirichlet problem. Commun. Pure Appl. Math. 62(3), 396–443 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Reese Harvey, F., Blaine Lawson, H.: The inhomogeneous Dirichlet problem for natural operators on manifolds. Ann. Inst. Fourier (Grenoble) 69(7), 3017–3064 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Reese Harvey, F., Blaine Lawson Jr., H.: Pseudoconvexity for the special Lagrangian potential equation. Calc. Var. Part. Differ. Equ. 60(1), 37 (2021)

  16. Ishii, H., Lions, P.-L.: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equ. 83(1), 26–78 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. Koike, S.: A beginner’s guide to the theory of viscosity solutions. MSJ Memoirs, vol. 13. Mathematical Society of Japan, Tokyo (2004)

  18. Krylov, N.V.: On the general notion of fully nonlinear second-order elliptic equations. Trans. Am. Math. Soc. 347(3), 857–895 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Luo, Y., Eberhard, A.: Comparison principles for viscosity solutions of elliptic equations via fuzzy sum rule. J. Math. Anal. Appl. 307(2), 736–752 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Supported by the Academy of Finland (grant SA13316965) and Aalto University. We thank the Reviewer for the comments on the earlier version of the manuscript, and for the suggestions on how to improve it.

Author information

Authors and Affiliations

  1. Frostavegen 1691, 7633, Frosta, Norway

    Karl K. Brustad

Authors
  1. Karl K. Brustad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karl K. Brustad.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brustad, K.K. On the comparison principle for second order elliptic equations without first and zeroth order terms. Nonlinear Differ. Equ. Appl. 30, 15 (2023). https://doi.org/10.1007/s00030-022-00819-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00030-022-00819-7

Keywords

Mathematics Subject Classification

Search

Navigation