Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the general notion of fully nonlinear second-order elliptic equations


Author: N. V. Krylov
Journal: Trans. Amer. Math. Soc. 347 (1995), 857-895
MSC: Primary 35J60; Secondary 35J65
DOI: https://doi.org/10.1090/S0002-9947-1995-1284912-8
MathSciNet review: 1284912
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The general notion of fully nonlinear second-order elliptic equation is given. Its relation to so-called Bellman equations is investigated. A general existence theorem for the equations like $ {P_m}({u_{{x^i}{x^j}}}) = \sum\nolimits_{k = 0}^{m - 1} {{c_k}(x){P_k}({u_{{x^i}{x^j}}})} $ is obtained as an example of an application of the general notion of fully nonlinear elliptic equations.


References [Enhancements On Off] (What's this?)

  • [1] M. F. Atiyah, R. Bott, and L. G[ill]rding, Lacunas for hyperbolic differential operators with constant coefficients, I, Acta Math. 124 (1970), 109-189. 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. Monge-Ampère equations, Comm. Pure Appl. Math. 37 (1984), 369-402. 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 Monge-Ampère, and uniformly elliptic, equations, Comm. Pure Appl. Math. 38 (1985), 209-252. 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), 261-301. 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), 1-67. MR 1118699 (92j:35050)
  • [8] L. G[ill]rding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959), 957-965. 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., North-Holland, Amsterdam, 1990. MR 1045639 (91a:32001)
  • [13] N.M. Ivochkina, A description of the stability cones generated by differential operators of Monge-Ampère type, Mat. Sb. 122 (1983), 265-275; English transl. Math. USSR-Sb. 50 (1985), 259-268. MR 717679 (85g:35043)
  • [14] -, Solution of the Dirichlet problem for curvature equation of order $ m$, Mat. Sb. 180 (1989), 867-887; English transl. Math. USSR-Sb. 67 (1990), 317-339. 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), 1029-1048; English transl. Math. USSR-Izv. 19 (1982), 297-313.
  • [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), 242-269; English transl. Math. USSR-Izv. 30 (1988), 217-244. MR 896996 (88h:35040)
  • [19] -, Unconditional solvability of the Bellman equation with constant coefficients in convex domains, Mat. Sb. 135 (1988), 297-311; English transl. Math. USSR-Sb. 63 (1989), 289-303. 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), 66-96; English transl. Math. USSR-Izv. 34 (1990), 65-95. 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), 69-72. 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), 1272-1287; English transl. Math. USSR-Izv. 33 (1989), 597-612. MR 984219 (90d:35104)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35J60, 35J65

Retrieve articles in all journals with MSC: 35J60, 35J65


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1284912-8
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society