Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds

Journal of Functional Analysis

Volumn 220, Issue 2, 2005, Pages 304-361

  Azagra, Daniel ^a   Ferrera, Juan ^a   López Mesas, Fernando ^a  

^a UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

Author keywords

Riemannian manifolds; Subdifferential

Indexed keywords

EID: 13644283271     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jfa.2004.10.008     Document Type: Article

References (47)

1. R. Abraham J.E. Marsden Foundations of Mechanics second ed. 1980 The Benjamin/Cummings Publishing Company, Inc. Reading, MA
2. D. Aussel J.-N. Corvellec M. Lassonde Mean value property and subdifferential criteria for lower semicontinuous functions Trans. Amer. Math. Soc. 347 1995 4147-4161
3. D. Azagra, Smooth negligibility and subdifferential calculus in Banach spaces, with applications, Ph.D. Thesis, Universidad Complutense de Madrid, October 1997.
4. D. Azagra M. Cepedello Boiso Uniform approximation of mappings by smooth mappings with no critical points on Hilbert manifolds Duke Math. J. 124 1 2004 47-66
5. D. Azagra R. Deville Subdifferential Rolle's and mean value inequality theorems Bull. Austral. Math. Soc. 56 1997 319-329
6. D. Azagra J. Gómez J.A. Jaramillo Rolle's theorem and negligibility of points in infinite-dimensional Banach espaces J. Math. Anal. Appl. 213 1997 487-495
7. D. Azagra M. Jiménez-Sevilla The failure of Rolle's theorem in infinite-dimensional Banach spaces J. Funct. Anal. 182 2001 207-226
8. G. Barles Solutions de viscosité des équations de Hamilton-Jacobi, Mathematics and Applications, Paris 1994 Springer Berlin
9. J. Bes, J. Ferrera, On a multidimensional version of Rolle's theorem, Publ. Dto. Análisis Mat. UCM, Ser. 1 no. 42, 1996-1997, pp. 21-27.
10. F.H. Clarke, Yu.S. Ledyaev, R.J. Stern, P.R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, vol. 178, 1998 Springer, Berlin.
11. M.G. Crandall L.C. Evans P.L. Lions Some properties of viscosity solutions to Hamilton-Jacobi equations Trans. Amer. Math. Soc. 282 1984 487-502
12. M.G. Crandall H. Ishii P.L. Lions Uniqueness of viscosity solutions revisited J. Math. Soc. Japan 39 1987 581-596
13. M.G. Crandall H. Ishii P.L. Lions User's guide to viscosity solutions of second order to fully nonlinear partial differential equations Bull. Amer. Math. Soc. 27 1992 1-67
14. M.G. Crandall P.L. Lions Solutions de viscosité non bornées des équations de Hamilton-Jacobi du premier ordre C. R. Acad. Sci. Paris 198 1981 217-220
15. M.G. Crandall P.L. Lions Viscosity solutions of Hamilton-Jacobi equations Trans. Amer. Math. Soc. 277 1983 1-42
16. M.G. Crandall P.L. Lions Condition d'unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre C. R. Acad. Sci. Paris 292 1984 183-186
17. M.G. Crandall, P.L. Lions, Hamilton-Jacobi equations in infinite dimensions, Part I: Uniqueness of viscosity solutions, J. Funct. Anal. 62 (1985) 379-396
18. M.G. Crandall, P.L. Lions, Hamilton-Jacobi equations in infinite dimensions, Part II: Existence of viscosity solutions, J. Funct. Anal. 65 (1986) 368-405
19. M.G. Crandall, P.L. Lions, Hamilton-Jacobi equations in infinite dimensions, Part III, J. Funct. Anal. 68 (1986) 214-247
20. M.G. Crandall, P.L. Lions Hamilton-Jacobi equations in infinite dimensions, Part IV: Ubounded linear terms, J. Funct. Anal. 90 (1990) 237-283
21. M.G. Crandall, P.L. Lions Hamilton-Jacobi equations in infinite dimensions, Part V: B-continuous solutions, J. Funct. Anal. 97 (1991) 417-465
22. M.G. Crandall P.L. Lions On existence and uniqueness of solutions of Hamilton-Jacobi equations Non. Anal. Theor. Math. Appl. 10 1986 353-370
23. M.G. Crandall P.L. Lions Unbounded viscosity solutions of Hamilton-Jacobi equations Illinois J. Math. 31 1987 665-688
24. M.G. Crandall P.L. Lions Solutions de viscosité pour les équations de Hamilton-Jacobi en dimension infinie intervenant dans le contrôle optimal de problèmes d'évolution C. R. Acad. Sci. Paris 305 1987 233-236
25. R. Deville Stability of subdifferentials of nonconvex functions in Banach spaces Set-Valued Anal. 2 1994 141-157
26. R. Deville A mean value theorem for the non differentiable mappings Serdica Math. J. 21 1995 59-66
27. R. Deville, Smooth variational principles and nonsmooth analysis in Banach spaces, in: F.H. Clarke, R.J. Stern (Eds.), Nonlinear Analysis, Differential Equations and Control, NATO Sci. Ser. C Math. Phys. Sci., vol. 528, Kluwer Academic Publishers, Dordrecht, 1999, pp. 369-405.
28. R. Deville E.El. Haddad The viscosity subdifferential of the sum of two functions in Banach spaces, I, First order case J. Convex Anal. 3 2 1996 295-308
29. R. Deville E.M. El Haddad The subdifferential of the sum of two functions in Banach spaces, II, Second order case Bull. Austral. Math. Soc. 51 1995 235-248
30. R. Deville N. Ghoussoub Perturbed minimization principles and applications in: W.B. Johnson J. Lindenstrauss (Eds.) Handbook of the Geometry of Banach Spaces vol. 1 2001 North-Holland Elsevier Science, Amsterdam 393-435
31. R. Deville, G. Godefroy, V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monographies and Surveys in Pure and Applied Mathematics, vol. 64, Pitman, London, 1993.
32. R. Deville M. Ivanov Smooth variational principles with constraints Arch. Math. 69 1997 418-426
33. I. Ekeland The Hopf-Rinow theorem in infinite dimension J. Differential Geom. 13 2 1978 287-301
34. I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1(3) (New series) (1979) 443-474.
35. H. Federer, Geometric measure theory, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer, New York, NY, 1969.
36. G. Godefroy Some remarks on subdifferential calculus Rev. Mat. Complutense 11 2 1998 269-279
37. R.E. Greene K. Shiohama Convex functions on complete noncompact manifolds: Topological structure Invent. Math. 63 1981 129-157
38. R.E. Greene K. Shiohama Convex functions on complete noncompact manifolds: Differentiable structure Ann. Sci. École. Norm. Sup. 4 série 14 1981 357-367
39. R.E. Greene H. Wu C∞ convex functions and manifolds of positive curvature Acta Math. 137 1976 209-245
40. R.E. Greene H. Wu C∞ approximations of convex, subharmonic, and plurisubharmonic functions Ann. Sci. École. Norm. Sup. 4 série 12 1979 47-84
41. E.M. El Haddad, Calcul sous-differentiel et solutions de viscosité des équations de Hamilton-Jacobi, Thèse doctorale présentée à l'Université Bordeaux I en 1994.
42. W. Klingenberg, Riemannian Geometry, Walter de Gruyter Studies in Mathematics, vol. 1, Walter de Gruyter, Berlin, New York, 1982.
43. I. Kupka Counterexample to the Morse-Sard theorem in the case of infinite-dimensional manifolds Proc. Amer. Math. Soc. 16 1965 954-957
44. S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics, vol. 191, Springer, New York, 1999.
45. J. Milnor Morse Theory, Ann. of Math. Studies 1963 Princeton University Press Princeton
46. R.R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics, vol. 1364, Springer, Berlin, New York, 1993.
47. D. Preiss Differentiability of Lipschitz functions on Banach spaces J. Funct. Anal. 91 1990 312-345