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  • Bulletin of the Australian Mathematical Society, Volume 39, Issue 3
  • June 1989, pp. 443-447

On the twice differentiability of viscosity solutions of nonlinear elliptic equations

  • Neil S. Trudinger (a1)
  • DOI: http://dx.doi.org/10.1017/S0004972700003361
  • Published online: 17 April 2009
Abstract

We prove, under very general structure conditions, that continuous viscosity subsolutions of nonlinear second-order elliptic equations possess second order superdifferentials almost everywhere. Consequently we deduce the twice differentiability almost everywhere of viscosity solutions. The main idea of the proof is the backwards use of the Aleksandrov maximum principle as invoked in a previous work of Nadirashvili on sequences of solutions of linear elliptic equations.

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[3]M.G. Crandall and P.-L. Lions , ‘Viscosity solutions of Hamilton-Jacobi equations’, Trans. Amer. Math. Soc. 277 (1983), 142.

[4]D. Gilbarg and N.S. Trudinger , Elliptic partial differential equations of the second order 2nd edition (Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983).

[7]R. Jensen , ‘The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations’, Arch. Rational Mech. Anal. 101 (1988), 127.

[11]P.L. Lions , ‘Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations II’, Comm. Partial Differential Equations 8 (1983), 12291276.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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