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A remark on Bony maximum principle


Author: P.-L. Lions
Journal: Proc. Amer. Math. Soc. 88 (1983), 503-508
MSC: Primary 35J65; Secondary 35B50
DOI: https://doi.org/10.1090/S0002-9939-1983-0699422-3
MathSciNet review: 699422
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Abstract: We extend a result due to J. M. Bony concerning a form of the classical maximum principle adapted to Sobolev spaces. We treat the case of the limiting exponent and show that the result is optimal. We give various applications to nonlinear elliptic partial differential equations.


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

  • [1] A. D. Alexandrov, Investigations on the maximum principle, Izv. Vysš. Učebn. Zaved Matematika. I, 5 (1958), 126-157; II, 3 (1959), 3-12; III, 5 (1959), 16-32; IV, 3 (1960), 3-15; V, 5 (1960), 16-26; VI, 1 (1961), 3-20. (Russian)
  • [2] -, Uniqueness conditions and estimates for the solution of the Dirichlet problem, Amer. Math. Soc. Transl. (2) 68 (1968), 89-119.
  • [3] -, Majorization of solutions of second-order linear equations, Amer. Math. Soc. Transl. (2) 68 (1968), 120-143.
  • [4] -, Majorants of solutions and uniqueness conditions for elliptic equations, Amer. Math. Soc. Transl. (2) 68 (1968), 144-161.
  • [5] -, The impossibility of general estimates for solutions and of uniqueness conditions for linear equations with norms weaker than in $ {L_n}$, Amer. Math. Soc. Transl. (2) 68 (1968), 162-168.
  • [6] -, Dirichlet's problem for the equation $ {\operatorname{Det}}\left\Vert {{z_{ij}}} \right\Vert = \Phi ({z_1}, \ldots ,{z_n},z,{x_1}, \ldots ,{x_n})$. I, Vestnik Leningrad Univ. Mat. Meh. Astronom. 13 (1958), 5-24. (Russian)
  • [7] H. Amann and M. G. Crandall, On some existence theorems for semilinear elliptic equations, Indiana Univ. Math. J. 27 (1978), 779-790. MR 503713 (80a:35047)
  • [8] J. M. Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris Ser. A 265 (1967), 333-336. MR 0223711 (36:6759)
  • [9] S. Y. Cheng and S. T. Yau, On the regularity of the Monge-Ampère equation $ \det ({\partial ^2}u/\partial {x_i}\partial {x_j}) = F(x,u)$, Comm. Pure Appl. Math. 30 (1977), 41-68. MR 0437805 (55:10727)
  • [10] M. G. Crandall, L. C. Evans and P. L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. (to appear). MR 732102 (86a:35031)
  • [11] M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. (to appear). MR 690039 (85g:35029)
  • [12] -, Condition d'unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre, C. R. Acad. Sci. Paris Ser. A 292 (1981), 183-186. MR 610314 (82c:49020)
  • [13] L. C. Evans, A convergence theorem for solutions of nonlinear second-order elliptic equations, Indiana Univ. Math. J. 27 (1978), 875-887. MR 503721 (80e:35023)
  • [14] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York, 1980. MR 567696 (81g:49013)
  • [15] N. V. Krylov, Controlled diffusion processes, Springer-Verlag, Berlin, 1980. MR 601776 (82a:60062)
  • [16] -, An inequality in the theory of stochastic integrals, Theory Probab. Appl. 17 (1972), 114-131.
  • [17] -, Some estimates of the probability density of a stochastic integral, Math. USSR-Izv. 8 (1974), 233-254.
  • [18] H. Lewy and G. Stampacchia, On the smoothness of superharmonics which solve a minimum problem, J. Analyse Math. 23 (1970), 227-236. MR 0271383 (42:6266)
  • [19] P. L. Lions, Resolution analytique des problèmes de Bellman-Dirichlet, Acta Math. 146 (1981), 151-166. MR 611381 (83c:49038)
  • [20] -, Optimal stochastic control of diffusion type processes and Hamilton-Jacobi-Bellman equations, Advances in Filtering and Optimal Stochastic Control (Eds., W. H. Fleming and L. Gorostiza), Springer, Berlin, 1982. MR 794517 (86h:93073)
  • [21] -, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations (preprint).
  • [22] -, Sur les équations de Monge-Ampère. III (in preparation).
  • [23] -, Une méthode nouvelle pour l'existence de solutions régulières de l'équations de Monge-A mpère, C. R. Acad. Sci. Paris Ser. A 293 (1981), 589-592. MR 647688 (83d:58025)
  • [24] -, Sur les équations de Monge-Ampère. I, Manuscripta Math. (to appear); II, Arch. Rational Mech. Anal. (to appear).
  • [25] A. V. Pogorelov, The Minkowski multidimensional problem, Wiley, New York, 1978. MR 0478079 (57:17572)
  • [26] C. Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl. 74 (1966), 15-30. MR 0214905 (35:5752)
  • [27] C. Pucci and G. Talenti, Elliptic (second-order) partial differential equations with measurable coefficients and approximating integral equations, Adv. in Math. 19 (1976), 48-105. MR 0419989 (54:8006)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0699422-3
Keywords: Sobolev spaces, maximum principle, second-order elliptic equations, viscosity solutions
Article copyright: © Copyright 1983 American Mathematical Society

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