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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A remark on Bony maximum principle
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by P.-L. Lions
Proc. Amer. Math. Soc. 88 (1983), 503-508
DOI: https://doi.org/10.1090/S0002-9939-1983-0699422-3

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
    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) —, Uniqueness conditions and estimates for the solution of the Dirichlet problem, Amer. Math. Soc. Transl. (2) 68 (1968), 89-119. —, Majorization of solutions of second-order linear equations, Amer. Math. Soc. Transl. (2) 68 (1968), 120-143. —, Majorants of solutions and uniqueness conditions for elliptic equations, Amer. Math. Soc. Transl. (2) 68 (1968), 144-161. —, 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. —, Dirichlet’s problem for the equation ${\operatorname {Det}}\left \| {{z_{ij}}} \right \| = \Phi ({z_1}, \ldots ,{z_n},z,{x_1}, \ldots ,{x_n})$. I, Vestnik Leningrad Univ. Mat. Meh. Astronom. 13 (1958), 5-24. (Russian)
  • Herbert Amann and Michael G. Crandall, On some existence theorems for semi-linear elliptic equations, Indiana Univ. Math. J. 27 (1978), no. 5, 779–790. MR 503713, DOI 10.1512/iumj.1978.27.27050
  • Jean-Michel Bony, Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A333–A336 (French). MR 223711
  • Shiu Yuen Cheng and Shing Tung Yau, On the regularity of the Monge-Ampère equation $\textrm {det}(\partial ^{2}u/\partial x_{i}\partial sx_{j})=F(x,u)$, Comm. Pure Appl. Math. 30 (1977), no. 1, 41–68. MR 437805, DOI 10.1002/cpa.3160300104
  • M. G. Crandall, L. C. Evans, and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), no. 2, 487–502. MR 732102, DOI 10.1090/S0002-9947-1984-0732102-X
  • Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. MR 690039, DOI 10.1090/S0002-9947-1983-0690039-8
  • Michael G. Crandall and Pierre-Louis Lions, Condition d’unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 3, 183–186 (French, with English summary). MR 610314
  • Lawrence C. Evans, A convergence theorem for solutions of nonlinear second-order elliptic equations, Indiana Univ. Math. J. 27 (1978), no. 5, 875–887. MR 503721, DOI 10.1512/iumj.1978.27.27059
  • David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696
  • N. V. Krylov, Controlled diffusion processes, Applications of Mathematics, vol. 14, Springer-Verlag, New York-Berlin, 1980. Translated from the Russian by A. B. Aries. MR 601776
  • —, An inequality in the theory of stochastic integrals, Theory Probab. Appl. 17 (1972), 114-131. —, Some estimates of the probability density of a stochastic integral, Math. USSR-Izv. 8 (1974), 233-254.
  • Hans Lewy and Guido Stampacchia, On the smoothness of superharmonics which solve a minimum problem, J. Analyse Math. 23 (1970), 227–236. MR 271383, DOI 10.1007/BF02795502
  • P.-L. Lions, Résolution analytique des problèmes de Bellman-Dirichlet, Acta Math. 146 (1981), no. 3-4, 151–166 (French). MR 611381, DOI 10.1007/BF02392461
  • P.-L. Lions, Optimal stochastic control of diffusion type processes and Hamilton-Jacobi-Bellman equations, Advances in filtering and optimal stochastic control (Cocoyoc, 1982) Lect. Notes Control Inf. Sci., vol. 42, Springer, Berlin, 1982, pp. 199–215. MR 794517, DOI 10.1007/BFb0004539
  • —, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations (preprint). —, Sur les équations de Monge-Ampère. III (in preparation).
  • Pierre-Louis Lions, Une méthode nouvelle pour l’existence de solutions régulières de l’équation de Monge-Ampère réelle, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 12, 589–592 (French, with English summary). MR 647688
  • —, Sur les équations de Monge-Ampère. I, Manuscripta Math. (to appear); II, Arch. Rational Mech. Anal. (to appear).
  • Aleksey Vasil′yevich Pogorelov, The Minkowski multidimensional problem, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto-London, 1978. Translated from the Russian by Vladimir Oliker; Introduction by Louis Nirenberg. MR 0478079
  • Carlo Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl. (4) 74 (1966), 15–30 (Italian, with English summary). MR 214905, DOI 10.1007/BF02416445
  • Carlo Pucci and Giorgio Talenti, Elliptic (second-order) partial differential equations with measurable coefficients and approximating integral equations, Advances in Math. 19 (1976), no. 1, 48–105. MR 419989, DOI 10.1016/0001-8708(76)90022-0
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Bibliographic Information
  • © Copyright 1983 American Mathematical Society
  • 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