The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations Authors
Cite this article as: Jensen, R. Arch. Rational Mech. Anal. (1988) 101: 1. doi:10.1007/BF00281780 Abstract
We prove that viscosity solutions in
W 1,∞ of the second order, fully nonlinear, equation F( D 2 u, Du, u) = 0 are unique when (i) F is degenerate elliptic and decreasing in u or (ii) F is uniformly elliptic and nonincreasing in u. We do not assume that F is convex. The method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions, respectively. This method is completely different from that used in Lions [8, 9] for second order problems with F convex in D 2 u and from that used by Crandall & Lions  and Crandall, Evans & Lions  for fully nonlinear first order problems. Communicated by C. M. Dafermos
The research reported here was supported in part by grants from the Alfred P. Sloan Foundation and the National Science Foundation.
Bony, Principe du maximum dans les espaces de Sobolev. C. R. Acad. Sci. Paris 265 (1967), 333–336.
Crandall, L. C. Evans, & P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984), 487–502.
Crandall & P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983), 1–42.
Evans, A convergence theorem for solutions of nonlinear second order elliptic equations. Indiana Univ. Math. J. 27 (1978), 875–887.
Evans, On solving certain nonlinear partial differential equations by accretive operator methods. Israel J. Math. 36 (1980), 225–247.
Evans, Classical solutions of fully nonlinear, convex, second order elliptic equations. Comm. Pure Appl. Math. 25 (1982), 333–363.
Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 1. Dynamic Programming Principle and applications. Comm. Partial Differential Equations 8 (1983), 1101–1134.
Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 2. Viscosity solutions and uniqueness. Comm. Partial Differential Equations 8 (1983), 1229–1276.
Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 3. Regularity of the optimal cost function. Collège de France Seminar, Vol. V, Pitman, London, 1983.
Pucci, Limitazioni per soluzioni di equazioni ellitiche. Ann. Mat. Pura Appl. 74 (1966), 15–30.
Stein, Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, New Jersey, 1970. Copyright information
© Springer-Verlag GmbH & Co 1988