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Sur les equations de Monge-Ampere. I

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Abstract

In this paper we study the real Monge-Arapère equations: det(D2u)= f(x) in 0, u convex in 0, u=0 on ∂0, and we introduce a new method for solving these equations which enables us to show the existence of regular solutions. This method uses only p.d.e. techniques and does not use any geometrical results. Furthermore, it enables us to solve quasilinear Monge-Ampère equations.

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Bibliographie

  1. ALEXANBROV, A.D.: Dirichlet's problem for the equation Det¦¦zij¦¦=φ(z1,..., zn, z,x1,...,xn), I, Vestnik Leningrad Univ. Ser. Mat. Mekh. Astr.13 (1958), pp. 5–24 (en Russe)

    Google Scholar 

  2. AUBIN, T.: Equations du type de Monge-Ampère sur les variétés kählériennes compactes, Comptes-Rendus Paris,283 (1976), pp. 119–121

    Google Scholar 

  3. AUBIN, T.: Equations du type de Monge-Ampére sur les variétés kahlériennes compactes, Bull. Sc. Math.102 (1978), pp. 63–95

    Google Scholar 

  4. AUBIN, T.: Equations de Monge-Ampère réelles, J. Funct. Anal.41 (1981), pp. 354–377

    Google Scholar 

  5. BEDFORD, E.: Extremal plurisubharmonic functions for certain domains in C2, Ind. Univ. Math. J.,28 (1979), pp. 613–626

    Google Scholar 

  6. BEDFORD, E.: An example of a plurisubharmonic measure on the unit ball in C2. Preprint

  7. BEDFORD, E.: Stability of the envelope of holomorphy of two paraboloids and the degenerate Monge--Arnpére equation. Preprint

  8. BEDFORD, E. et TAYLOR, B.A.: The Dirichlet problem for a complex Monge-Ampère equation. Inventiones Math.,37 (1976), pp. 1–44

    Google Scholar 

  9. BEDFORD, E. et TAYLOR, B.A.: Variational properties of the complex Monge-Ampère equation, I. Dirichlet Principle. Duke Math. J.,45 (1978), pp. 375–403

    Google Scholar 

  10. BEDFORD, E. et TAYLOR, B.A.: Variational properties of the complex Monge-Ampère equation, II. Intrinsic Norms. Amer. J. Math.,101 (1979), pp. 1131–1166

    Google Scholar 

  11. BONY, J.M.: Principe du maximum dans les espaces de Sobolev, Comptes-Rendus Paris,265 (1967), pp. 333–336

    Google Scholar 

  12. CALABI, E.: Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. Michigan Math. J.,5 (1958), pp. 105–126

    Google Scholar 

  13. CHENG, S.Y. et YAU, S.T.: On the regularity of the Monge-Ampère equation det(∂2u/∂x-∂xj)=F(x,u). Comm. Pure Appl. Math.,30 (1977), pp. 41–68

    Google Scholar 

  14. CHENG, S.Y. et YAU, S.T.: On the regularity of the solution of the n-dimensional Minkowski problem. Comm. Pure Appl. Math

  15. CHENG, S.Y. et Yau, S.T.: On the existence of a complete Kahler metric on non-compact complex manifolds and the regularity of Fefferman's equation. Comm. Pure Appl. Math.33 (1980), pp-507–544

    Google Scholar 

  16. CHERRIER, P.: Etude d'une équation du type Monge--Ampère sur les variétés kählériennes compactes. Comptes-Rendus Paris,295 (1981), pp. 277–279

    Google Scholar 

  17. CRANDALL, M.G. et LIONS, P.L.: Condition d'unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre. Comptes-Rendus Paris,292 (1981), pp. 183–186

    Google Scholar 

  18. CRANDALL, M.G. et LIONS, P.L.: Viscosity solutions of Hamilton-Jacobi équations. A parattre aux Trans. Amer. Math. Soc. (1982)

  19. DELANOE, P.: Equations du type Monge-Ampère sur les variétés Riemanniennes compactes, I. J. Funct. Anal.,40 (1981), pp. 358–386

    Google Scholar 

  20. DELANOE, P.: Equations du type Monge-Ampère sur les variétés Riemanniennes compactes, II. J. Funct. Anal.,41 (1981), pp. 341–353

    Google Scholar 

  21. EVANS, L.C.: A convergence theorem for solutions of nonlinear second order elliptic equations. Ind. Univ. Math. J.

  22. EVANS, L.C.: Classical solutions of fully nonlinear, convex second-order elliptic equations, Comm. Pure Appl. Math. (1982)

  23. EVANS, L.C.: Classical solutions of the Hamilton--Jacobi-Bellman equation for uniformly elliptic operators. Preprint

  24. EVANS, L.C. et FRIEDMAN, A.: Optimal stochastic switching and the Dirichlet problem for the Bellman equation. Trans. Amer. Math. Soc.,253 (1979), pp. 365–389

    Google Scholar 

  25. GAVEAU, B.: Méthode de contrôle optimal en analyse complexe. I. Résolution d'équations de Monge-Ampère. J. Funct. Anal.,25 (1977), pp. 391–411

    Google Scholar 

  26. GAVEAU, B.: Méthode de contrôle optimal en analyse complexe. II. Equation de Monge-Ampère dans certains domaines faiblement pseudoconvexes. Bull. Sc. Math.,102 (1978), pp. 101–128

    Google Scholar 

  27. KRYLOV, N.V.: On control of the solution of a stochastic integral equation with degeneration. Math. USSR Izv.,6 (1972), pp. 249–262

    Google Scholar 

  28. KRYLOV, N.V.: Controlled diffusion processes. Springer-Verlag, Berlin (1980)

    Google Scholar 

  29. LIONS, P.L.: Control of diffusion processes in RN. Comm. Pure Appl. Math.34 (1981), pp. 121–147

    Google Scholar 

  30. LIONS, P.L.: Sur les équations de Monge-Ampère. II. A paraître aux Arch. Rat. Mech. Anal

  31. LIONS, P.L.: Résolution analytique des problèmes de Bellman-Dirichlet. Acta Mathematica,146 (1981), pp. 151–166

    Google Scholar 

  32. LIONS, P.L.: Equations de Hamilton-Jacobi-Bellman. In Sèminaire Gaulaouic-Schwartz 1979–1980, Ecole Polytechnique, Paris

  33. LIONS, P.L.: Optimal stochastic control and Hamilton--Jacobi-Bellman equations. A paraître dans Mathematical Control Theory, Banach Center Publications, Varsovie

  34. LIONS, P.L.: Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, I, II, III (en préparation); voir aussi Comptes-Rendus Paris,289 (1979), pp. 329–332

    Google Scholar 

  35. LIONS, P.L.: Bifurcation and optimal stochastic control. A paraître au Nonlinear Anal. T.M.A; voir aussi MRC report,Univ. Wisconsin-Madison (1982)

  36. LIONS, P.L.: Une méthode nouvelle pour l'existence de solutions régulières de l'équation de Monge-Ampère. Comptes-Rendus Paris,293 (1981), pp. 589–592

    Google Scholar 

  37. LIONS, P.L.: Résolution de problèmes quasilinèaires, Arch. Rat. Mech. Anal.,74 (1980), pp. 335–354

    Google Scholar 

  38. LIONS, P.L. et MENALDI, J.L.: Problème de Bellman avec le contrôle dans les coefficients de plus haut degré. Comptes-Rendus Paris,287 (1978), pp. 409–412

    Google Scholar 

  39. LIONS, P.L. et MENALDI, J.L.: Optimal control of stochastic integrals and Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. (1982)

  40. NIRENBERG, L.: Monge-Ampére equations and some associated problems in geometry. Proceedings of the International Congress of Mathematicians, Vancouver, 1974

  41. POGORELOV, A.V.: On a regular solution of the n-dimensional Minkowski problem. Soviet. Math. Dokl.12 (1971), pp. 1192–1196

    Google Scholar 

  42. POGORELOV, A.V.: The Minkowski multidimensional problem. J. Wiley, New York (1978)

    Google Scholar 

  43. POGORELOV, A.V.: The Dirichlet problem for the n-dimensional analogue of the Monge-Ampère equation. Soviet. Math. Dokl.12 (1971), pp. 1727–1731

    Google Scholar 

  44. POGORELOV, A.V.: On the regularity of generalized solutions of the equation det(∂2u/∂xi ∂xj)= ϕ(x1,...,xn) > 0. Soviet Math. Dokl.12 (1971), p. 1436–1440

    Google Scholar 

  45. PUCCI, C.: Limitazioni per soluzioni di equationi ellitiche. Ann. Mat. Pura Appl.,74 (1966), pp. 15–30

    Google Scholar 

  46. RAUCH, J. et TAYLOR, B.A.: The Dirichlet problem for the multidimensional Monge-Ampère equation. Rocky Mountain J. Math.,17 (1977), pp. 345–354

    Google Scholar 

  47. TRUDINGER, N.S.: Elliptic equations in non-divergence form. Preprint

  48. YAU, S.T.: On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampère equation, I. Comm. Pure Appl. Math.,31 (1978), pp. 339–411

    Google Scholar 

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  1. Ceremade Université Paris IX-Dauphine, Place de Lattre de Tassigny, F-75775, Paris Cedex 16

    Pierre-Louis Lions

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Lions, PL. Sur les equations de Monge-Ampere. I. Manuscripta Math 41, 1–43 (1983). https://doi.org/10.1007/BF01165928

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