On the general notion of fully nonlinear secondorder elliptic equations
Author:
N. V. Krylov
Journal:
Trans. Amer. Math. Soc. 347 (1995), 857895
MSC:
Primary 35J60; Secondary 35J65
MathSciNet review:
1284912
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Abstract: The general notion of fully nonlinear secondorder elliptic equation is given. Its relation to socalled Bellman equations is investigated. A general existence theorem for the equations like is obtained as an example of an application of the general notion of fully nonlinear elliptic equations.
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 [1]
 M. F. Atiyah, R. Bott, and L. G[ill]rding, Lacunas for hyperbolic differential operators with constant coefficients, I, Acta Math. 124 (1970), 109189. MR 0470499 (57:10252a)
 [2]
 I. Ia. Bakelman, Geometricheskie metody reshenia ellipticheskikh uravnenni, "Nauka", Moscow, 1965. (Russian) MR 0194727 (33:2933)
 [3]
 L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, I. MongeAmpère equations, Comm. Pure Appl. Math. 37 (1984), 369402. MR 739925 (87f:35096)
 [4]
 L. Caffarelli, J. J. Kohn, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, II. Complex MongeAmpère, and uniformly elliptic, equations, Comm. Pure Appl. Math. 38 (1985), 209252. MR 780073 (87f:35097)
 [5]
 L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), 261301. MR 806416 (87f:35098)
 [6]
 R. Courant, Methods of mathematical physics, Vol. 2, Interscience, New York, 1966.
 [7]
 M. G. Crandall, H. Ishii, and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 167. MR 1118699 (92j:35050)
 [8]
 L. G[ill]rding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959), 957965. MR 0113978 (22:4809)
 [9]
 D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer Verlag, Berlin, 1983. MR 737190 (86c:35035)
 [10]
 G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, 2nd ed., Cambridge University Press, Cambridge, 1952. MR 0046395 (13:727e)
 [11]
 L. Hörmander, The analysis of linear partial differential operators, Vols. 1, 2, Springer, New York, 1990.
 [12]
 , An introduction to complex analysis in several variables, 3rd ed., NorthHolland, Amsterdam, 1990. MR 1045639 (91a:32001)
 [13]
 N.M. Ivochkina, A description of the stability cones generated by differential operators of MongeAmpère type, Mat. Sb. 122 (1983), 265275; English transl. Math. USSRSb. 50 (1985), 259268. MR 717679 (85g:35043)
 [14]
 , Solution of the Dirichlet problem for curvature equation of order , Mat. Sb. 180 (1989), 867887; English transl. Math. USSRSb. 67 (1990), 317339. MR 1014618 (91a:35065)
 [15]
 N.V. Krylov, Controlled diffusion process, "Nauka", Moscow, 1977; English transl. by Springer, New York, 1980. MR 601776 (82a:60062)
 [16]
 , On control of a diffusion process up to the time of first exit from a region, Izv. Akad. Nauk Armyan. SSR, Ser. Math. (1981), 10291048; English transl. Math. USSRIzv. 19 (1982), 297313.
 [17]
 , Nonlinear elliptic and parabolic equations of second order, "Nauka", Moscow, 1985; English transl. by Reidel, Dordrecht, 1987.
 [18]
 , On the first boundary value problem for nonlinear degenerate elliptic equations, Izv. Akad. Nauk Armyan. SSR, Ser. Math., 51 (1987), 242269; English transl. Math. USSRIzv. 30 (1988), 217244. MR 896996 (88h:35040)
 [19]
 , Unconditional solvability of the Bellman equation with constant coefficients in convex domains, Mat. Sb. 135 (1988), 297311; English transl. Math. USSRSb. 63 (1989), 289303. MR 937642 (89h:35093)
 [20]
 , Smoothness of the payoff function for a controllable process in a domain, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 53 (1989), 6696; English transl. Math. USSRIzv. 34 (1990), 6595. MR 992979 (90f:93040)
 [21]
 M. Marcus and H. Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Boston, 1964. MR 0162808 (29:112)
 [22]
 W. Nuij, A note on hyperbolic polynomials, Math. Scand. 23 (1968), 6972. MR 0250128 (40:3368)
 [23]
 M.V. Safonov, On the classical solution of nonlinear elliptic equations of second order, Izv. Akad. Nauk Armyan. SSR. Math. 52 (1988), 12721287; English transl. Math. USSRIzv. 33 (1989), 597612. MR 984219 (90d:35104)
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DOI:
http://dx.doi.org/10.1090/S00029947199512849128
PII:
S 00029947(1995)12849128
Article copyright:
© Copyright 1995
American Mathematical Society
