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Dirichlet Problems for Fully Nonlinear Equations with “Subquadratic” Hamiltonians

  • Isabeau BirindelliEmail author
  • Françoise Demengel
  • Fabiana Leoni
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 33)

Abstract

For a class of fully nonlinear equations having second order operators which may be singular or degenerate when the gradient of the solutions vanishes, and having first order terms with power growth, we prove the existence and uniqueness of suitably defined viscosity solutions of Dirichlet problems and we further show that it is a Lipschitz continuous function.

Keywords

Inhomogenous equations Degenerate and singular fully nonlinear elliptic PDE Dirichlet problem 

2010Mathematical Subject Classification

35J70 35J75 

Notes

Acknowledgements

Part of this work has been done while the first and third authors were visiting the University of Cergy-Pontoise and the second one was visiting Sapienza University of Rome, supported by INDAM-GNAMPA and Laboratoire AGM Research Center in Mathematics of the University of Cergy-Pontoise.

References

  1. 1.
    A. Attouchi, E. Ruosteenoja, Remarks on regularity for p-Laplacian type equations in non-divergence form. J. Differ. Equ. 265, 1922–1961 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    G. Barles, F. Da Lio, On the generalized Dirichlet problem for viscous Hamilton–Jacobi equations. J. Math. Pures Appl. (9) 83(1), 53–75 (2004)Google Scholar
  3. 3.
    G. Barles, E. Chasseigne, C. Imbert, Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. 13, 1–26 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    I. Birindelli, F. Demengel, First eigenvalue and maximum principle for fully nonlinear singular operators. Adv. Differ. Equ. 11(1), 91–119 (2006)MathSciNetzbMATHGoogle Scholar
  5. 5.
    I. Birindelli, F. Demengel, \({\mathcal C}^{1,\beta }\) regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations. ESAIM Control Optim. Calc. Var. 20(40), 1009–1024 (2014)Google Scholar
  6. 6.
    I. Birindelli, F. Demengel, F. Leoni, Ergodic pairs for singular or degenerates fully nonlinear operators. ESAIM Control Optim. Calc. Var. (forthcoming).  https://doi.org/10.1051/cocv/2018070
  7. 7.
    I. Capuzzo Dolcetta, F. Leoni, A. Porretta, Hölder’s estimates for degenerate elliptic equations with coercive Hamiltonian. Trans. Am. Math. Soc. 362(9), 4511–4536 (2010)CrossRefGoogle Scholar
  8. 8.
    M.G. Crandall, H. Ishii, P.L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27(1), 1–67 (1992)Google Scholar
  9. 9.
    C. Imbert, L. Silvestre, C 1, α regularity of solutions of some degenerate fully nonlinear elliptic equations. Adv. Math. 233, 196–206 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    H. Ishii, Viscosity solutions of nonlinear partial differential equations. Sugaku Expositions, 144–151 (1996)Google Scholar
  11. 11.
    H. Ishii, P.L. Lions, Viscosity solutions of fully-nonlinear second order elliptic partial differential equations. J. Differ. Equ. 83, 26–78 (1990)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Isabeau Birindelli
    • 1
    Email author
  • Françoise Demengel
    • 2
  • Fabiana Leoni
    • 1
  1. 1.Dipartimento di MatematicaSapienza Università di RomaRomeItaly
  2. 2.Département de MathématiquesUniversité de Cergy-PontoiseCergy-PontoiseFrance

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