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An Asymptotic Mean Value Formula for Eigenvalues of the Hessian Related to Concave/Convex Envelopes

Abstract

In this paper we characterize viscosity solutions to the PDE λj(D2u) = 0 by means of the asymptotic mean value formula

$$ u(x) = \underset{\dim(S)=j}{\min} \underset{v\in S, |v|=1}{\max} \left\{\frac{1}{2} u (x + \epsilon v) + \frac{1}{2} u (x - \epsilon v)\right\} + o(\epsilon^{2}), $$

as 𝜖 → 0, that holds in the viscosity sense. Here, λ1(D2u) ≤⋯ ≤ λN(D2u) are the ordered eigenvalues of the Hessian D2u. This equation is related to optimal concave/convex envelopes.

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References

  1. Alexandroff, A.D.: Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it. Leningr. State Univ. Ann. Math. Ser. 6, 3–35 (1939)

    MathSciNet  Google Scholar 

  2. Birindelli, I., Galise, G., Leoni, F.: Liouville theorems for a family of very degenerate elliptic non lineal operators. Nonlinear Anal. 161, 198–211 (2017)

    MathSciNet  Article  Google Scholar 

  3. Birindelli, I., Galise, G., Ishii, I.: A family of degenerate elliptic operators: maximum principle and its consequences. Ann. Inst. H. Poincaré Anal. Non Linéaire 35, 417–441 (2018)

    MathSciNet  Article  Google Scholar 

  4. Blanc, P., Rossi, J.D.: Games for eigenvalues of the Hessian and concave/convex envelopes. J. Math. Pures Appl. 127, 192–215 (2019)

    MathSciNet  Article  Google Scholar 

  5. Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second-order elliptic equations, III : functions of the eigenvalues of the Hessian. Acta Math. 155, 261–301 (1985)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  7. Harvey, F.R., Lawson, H.B. Jr: Dirichlet duality and the nonlinear Dirichlet problem. Commun. Pure Appl. Math. 62, 396–443 (2009)

    MathSciNet  Article  Google Scholar 

  8. Harvey, F.R., Lawson, H.B. Jr: p-convexity, p-plurisubharmonicity and the Levi problem. Indiana Univ. Math. J. 62, 149–169 (2013)

    MathSciNet  Article  Google Scholar 

  9. Lindqvist, P., Manfredi, J.: On the mean value property for the p-Laplace equation in the plane. Proc. Amer. Math. Soc. 144, 143–149 (2016)

    MathSciNet  Article  Google Scholar 

  10. Manfredi, J.J., Parviainen, M., Rossi, J.D.: An asymptotic mean value characterization for p-harmonic functions. Proc. Amer. Math. Soc. 138, 881–889 (2010)

    MathSciNet  Article  Google Scholar 

  11. Oberman, A.M., Silvestre, L.: The Dirichlet problem for the convex envelope. Trans. Amer. Math. Soc. 363, 5871–5886 (2011)

    MathSciNet  Article  Google Scholar 

  12. Sha, J.-P.: Handlebodies and p-convexity. J. Differ. Geom. 25, 353–361 (1987)

    MathSciNet  Article  Google Scholar 

  13. Wu, H.: Manifolds of partially positive curvature. Indiana Univ. Math. J. 36, 525–548 (1987)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors are partially supported by CONICET grant PIP GI No 11220150100036CO (Argentina), by UBACyT grant 20020160100155BA (Argentina) and by MINECO MTM2015-70227-P (Spain).

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Correspondence to Julio D. Rossi.

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To Marco Antonio López on the occasion of his 70th birthday, with our best wishes.

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Blanc, P., Rossi, J.D. An Asymptotic Mean Value Formula for Eigenvalues of the Hessian Related to Concave/Convex Envelopes. Vietnam J. Math. 48, 335–344 (2020). https://doi.org/10.1007/s10013-020-00385-4

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  • DOI: https://doi.org/10.1007/s10013-020-00385-4

Keywords

  • Eigenvalues of the Hessian
  • Concave/convex envelopes
  • Mean value properties

Mathematics Subject Classification (2010)

  • 35D40
  • 35J25
  • 26B25