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Hölder gradient estimates for fully nonlinear elliptic equations

Published online by Cambridge University Press:  14 November 2011

Neil S. Trudinger
Neil S. Trudinger
Affiliation:
Max-Planck-Institut für Mathematik, Gottfried-Claren-Straβe 26, D-5300 Bonn 3, West Germany; Centre for Mathematical Analysis, Australian National University, Canberra G.P.O. Box 4, A.C.T. 2600, Australia
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Synopsis

In this paper we prove interior and global Hölder estimates for Lipschitz viscosity solutions of second order, nonlinear, uniformly elliptic equations. The smoothness hypotheses on the operators are more general than previously considered for classical solutions, so that our estimates are also new in this case and readily extend to embrace obstacle problems. In particular Isaac's equations of stochastic differential game theory constitute a special case of our results, and moreover our techniques, in combination with recent existence theorems of Ishii, lead to existence theorems for continuously differentiable viscosity solutions of the uniformly elliptic Isaac's equation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 108 , Issue 1-2 , 1988 , pp. 57 - 65
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Gilbarg, D. and Trudinger, N. S.Elliptic partial differential equations of second order, 2nd edn. (Berlin: Springer, 1983).Google Scholar
2Ishii, H.Perrons method for Hamilton-Jacobi equations. Duke Math. J. 55 (1987), 369–384.CrossRefGoogle Scholar
3Ishii, H. A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations (preprint).Google Scholar
4Ishii, H. On uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic PDE's (preprint).Google Scholar
5Jensen, R. The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations (to appear).Google Scholar
6Jensen, R., Lions, P.-L. and Souganidis, P. E. A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations. Proc. Amer. Math. Soc. (to appear).Google Scholar
7Krylov, N.V.Boundedly nonhomogeneous elliptic and parabolic equations in a domain. Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), 75–108; Math. USSR Izv. 21 (1984), 67-98.Google Scholar
8Krylov, N. V.. On derivative bounds for solutions of nonlinear parabolic equations. Dokl. Akad. Nauk UzSSR 274 (1983), 21–25.Google Scholar
9Krylov, N. V. and Safonov, M. V.. Certain properties of solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk. SSSR Ser. Mat. 40 (1980), 161–175; Math. USSR Izv. 16 (1981), 151-164.Google Scholar
10Ladyzhenskaya, O. A. and Ural'tseva, N. N.. Linear and quasilinear elliptic equations (Moscow: Izdat. Nauka, 1964); (New York: Academic Press, 1968).Google Scholar
11Lieberman, G. M.. The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary values. Comm. Partial Differential Equations 11 (1986), 167–229.CrossRefGoogle Scholar
12Lieberman, G. M.. Global regularity of solutions of nonlinear second order elliptic and parabolic differential equations. Math. Z. 193 (1986), 331–346.CrossRefGoogle Scholar
13Lions, P.-L.. Optimal control of diffusion processes and Hamiltonian-Jacobi Bellman equations II. Comm. Partial Differential Equations 8 (1983), 1229–1276.Google Scholar
14Trudinger, N. S.. Local estimates for subsolutions and super-solutions of general second order elliptic quasilinear equations. Invent. Math. 61 (1980), 67–79.CrossRefGoogle Scholar
15Trudinger, N. S.. Fully nonlinear, uniformly elliptic equations under natural structure conditions. Trans. Amer. Math. Soc. 278 (1983), 751–769.CrossRefGoogle Scholar
16Trudinger, N. S.. Boundary value problems for fully nonlinear elliptic equations. Proc. Centre Math. Anal. Aust. Nat. Univ. 8 (1984), 65–83.Google Scholar
17Trudinger, N. S.. On an interpolation inequality and its application to nonlinear elliptic equations. Proc. Amer. Math. Soc. 95 (1985), 75–78.Google Scholar
18Trudinger, N. S.. Comparison principles and pointwise estimates for viscosity solutions of second order, elliptic equations Aust. Nat. Univ. Centre for Math. Anal. Research Report 45 (1987).Google Scholar
19Trudinger, N. S.. On regularity and existence of viscosity solutions of second order, fully nonlinear elliptic equations Aust. Nat. Univ. Centre for Math. Anal. Research Report 46 (1987).Google Scholar