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

Vietnam Journal of Mathematics volume 48, pages 335–344 (2020)Cite this article

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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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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Authors and Affiliations

  1. Departamento de Matemática, FCEyN, Universidad de Buenos Aires, and IMAS - CONICET, Pabellón I, Ciudad Universitaria (1428), Buenos Aires, Argentina

    Pablo Blanc & Julio D. Rossi

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  1. Pablo Blanc
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  2. Julio D. Rossi
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Correspondence to Julio D. Rossi.

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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