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Principal eigenvalues for k-Hessian operators by maximum principle methods

1 Dipartimento di Matematica “G. Castelnuovo”, Sapienza Università di Roma, P.le Aldo Moro 2, 00185 Roma, 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: Critical values in nonlinear pdes – Special Issue dedicated to Italo Capuzzo Dolcetta
Guest Editor: Fabiana Leoni
Link: www.aimspress.com/mine/article/5754/special-articles

Special Issues: Critical values in nonlinear pdes - Special Issue dedicated to Italo Capuzzo Dolcetta

For fully nonlinear $k$-Hessian operators on bounded strictly $(k-1)$-convex domains $\Omega$ of $\mathbb{R}^N$, a characterization of the principal eigenvalue associated to a $k$-convex and negative principal eigenfunction will be given as the supremum over values of a spectral parameter for which admissible viscosity supersolutions obey a minimum principle. The admissibility condition is phrased in terms of the natural closed convex cone $\Sigma_k \subset {\cal S}(N)$ which is an elliptic set in the sense of Krylov [23] which corresponds to using $k$-convex functions as admissibility constraints in the formulation of viscosity subsolutions and supersolutions. Moreover, the associated principal eigenfunction is constructed by an iterative viscosity solution technique, which exploits a compactness property which results from the establishment of a global Hölder estimate for the unique $k$-convex solutions of the approximating equations.
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Keywords maximum principles; comparison principles; principal eigenvalues; k-Hessian operators; k-convex functions; admissible viscosity solutions; elliptic sets

Citation: Isabeau Birindelli, Kevin R. Payne. Principal eigenvalues for k-Hessian operators by maximum principle methods. Mathematics in Engineering, 2021, 3(3): 1-37. doi: 10.3934/mine.2021021

References

  • 1. Berestycki H, Nirenberg L, Varadhan S (1994) The principle eigenvalue and maximum principle for second order elliptic operators in general domains. Commun Pure Appl Math 47: 47-92.    
  • 2. Birindelli I, Demengel F (2006) First eigenvalue and maximum principle for fully nonlinear singular operators. Adv Differential Equ 11: 91-119.
  • 3. Birindelli I, Demengel F (2007) Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Commun Pure Appl Anal 6: 335-366.    
  • 4. Birindelli I, Galise G, Ishii H (2018) A family of degenerate elliptic operators: maximum principle and its consequences. Ann I H Poincaré Anal Non Linéaire 35: 417-441.    
  • 5. Caffarelli L, Nirenberg L, Spruck J, (1985) The Dirichlet problem for nonlinear second-order elliptic equations III. Functions of the eigenvalues of the Hessian. Acta Math 155: 261-301.
  • 6. 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.    
  • 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. Crandall MG, Ishii H, Lions PL (1992) User's guide to viscosity solutions of second order partial differential equations. Bull Am Math Soc 27: 1-67.    
  • 9. Gårding L, (1959) An inequality for hyperbolic polynomials. J Math Mech 8: 957-965.
  • 10. Gilbarg D, Trudinger NS (1983) Elliptic Partial Differential Equations of Second Order, 2 Eds., Berlin: Springer-Verlag.
  • 11. Harvey FR, Lawson HB (2009) Dirichlet duality and the nonlinear Dirichlet problem. Commun Pure Appl Math 62: 396-443.    
  • 12. Harvey FR, Lawson HB (2010) Hyperbolic polynomials and the Dirichlet problem, Available from: https://arxiv.org/abs/0912.5220.
  • 13. Harvey FR, Lawson HB (2011) Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds. J Differential Geom 88: 395-482.    
  • 14. Harvey FR, Lawson HB (2013) Gårding's theory of hyperbolic polynomials. Commun Pure Appl Math 66: 1102-1128.    
  • 15. Harvey FR, Lawson HB (2018) Tangents to subsolutions: existence and uniqueness, Part I. Ann Fac Sci Toulouse Math Ser 6 27: 777-848.    
  • 16. Harvey FR, Lawson HB (2018) The inhomogeneous Dirichlet Problem for natural operators on manifolds, Available from: https://arxiv.org/abs/1805.11121.
  • 17. Ishii H, Lions PL (1990) Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J Differential Equ 83: 26-78.    
  • 18. Lin M, Trudinger NS (1994) On some inequalities for elementary symmetric functions. Bull Aust Math Soc 50: 317-326.    
  • 19. Ivochkina NM (1981) The integral method of barrier functions and the Dirichlet problem for equations with operators of the Monge-Ampère type. Math USSR-Sb 29: 179-192.
  • 20. Ivochkina NM (1985) Description of cones of stability generated by differential operators of Monge-Ampère type. Math USSR-Sb 50: 259-268.    
  • 21. Korevaar NJ (1987) A priori bounds for solutions to elliptic Weingarten equations. Ann I H Poicaré Anal Nonlinéare 4: 405-421.
  • 22. Jacobsen J (1999) Global bifurcation problems associated with k-Hessian operators. Topol Method Nonl Anal 14: 81-130.    
  • 23. Krylov NV (1995) On the general notion of fully nonlinear second-order elliptic equations. T Am Math Soc 347: 857-895.    
  • 24. Labutin DA (2002) Potential estimates for a class of fully nonlinear elliptic equations. Duke Math J 111: 1-49.    
  • 25. Lieberman GW (1996) Second Order Parabolic Differential Equations, 2 Eds., Singapore: World Scientific Publishing Co Pte Ltd.
  • 26. Lions PL (1985) Two remarks in Monge-Ampère equations. Ann Mat Pura Appl 142: 263-275.    
  • 27. Trudinger NS (1995) On the Dirichlet problem for Hessian equations. Acta Math 175: 151-164.    
  • 28. Trudinger NS (1997) Weak solutions of Hessian equations. Commun Part Diff Equ 22: 1251-1261.
  • 29. Trudinger NS, Wang XJ (1997) Hessian measures I. Topol Method Nonl Anal 10: 225-239.    
  • 30. Trudinger NS, Wang XJ (1999) Hessian measures II. Ann Math 150: 579-604.    
  • 31. Urbas JIE (1999) On the existence of nonclassical solutions for two classes of fully nonlinear elliptic equations. Indiana Univ Math J 39: 335-382.
  • 32. Wang XJ (1995) A class of fully nonlinear elliptic equations and related functionals. Indiana U Math J 43: 25-54.
  • 33. Wang XJ (2009) The k-Hessian equation, In: Geometric Analysis and PDEs, Berlin: SpringerVerlag, 177-252.

 

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