Abstract

We present a formula for the viscosity subdifferential of the sum of two uniformly continuous functions on smooth Banach spaces. This formula is deduced from a new variational principle with constraints. We obtain as a consequence a weak form of Preiss’ theorem for uniformly continuous functions. We use these results to give simple proofs of some uniqueness results of viscosity solutions of Hamilton-Jacobi equations and we show how singlevaluedness of the associated Hamilton-Jacobi operators is related to the geometry of Banach spaces.

Author information

Contact details are reproduced from the original publication and may be historical.

E. El Haddad

Laboratoire de Mathématiques, Université de Franche-Comté, Route de Gray, 25030 Besancon, France.

R. Deville

Laboratoire de Mathématiques, Université Bordeaux I, 351, cours de la Libération, 33400 Talence, France.

References

  1. A. D. Ioffe: On subdifferentiability spaces, Ann. New York Acad. Sci. 410 (1983) 107–119.https://doi.org/10.1111/j.1749-6632.1983.tb23308.xGoogle Scholar
  2. M. Fabian: Subdifferentials, local ε-supports and Asplund spaces, J. London Math. Soc. 34 (1986) 568–576.https://doi.org/10.1112/jlms/s2-34.3.568Google Scholar
  3. M. Fabian: Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss, Acta Univ. Carolinae 30 (1989) 51–56.Google Scholar
  4. R. Deville, G. Godefroy and V. Zizler: A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, J. Funct. Anal. 111 (1993) 197–212.https://doi.org/10.1006/jfan.1993.1009Google Scholar
  5. J. M. Borwein, D. Preiss: A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987) 517–527.https://doi.org/10.1090/S0002-9947-1987-0902782-7Google Scholar
  6. M. Fabian, P. Hajek and J. Vanderwerff: variations on the smooth variational principle, to appear.Google Scholar
  7. R. Deville, G. Godefroy and V. Zizler: Smoothness and renormings in Banach spaces, Pitman Monographs in Mathematics 64, Longman Scientific and Technical, 1993.Google Scholar
  8. A. D. Ioffe: Proximal analysis and approximate subdifferentials, J. London Math. Soc. 41 (1990) 175–192.https://doi.org/10.1112/jlms/s2-41.1.175Google Scholar
  9. D. Preiss: Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal. 91 (1990) 312–345.https://doi.org/10.1016/0022-1236(90)90147-DGoogle Scholar
  10. M. Fabian: On a dual locally uniformly rotund norm on a dual Vasàk space, Studia Math. 101 (1991) 69–81.https://doi.org/10.4064/sm-101-1-69-81Google Scholar
  11. M. Fabian: Each weakly countably determined Asplund space admits a Fréchet differentiable norm, Bull. Austr. Math. Soc. 36 (1987) 367–374.https://doi.org/10.1017/S000497270000366XGoogle Scholar
  12. M. Talagrand: Renormages de quelques C(K), Israel J. Math. 54 (1986) 327–334.https://doi.org/10.1007/BF02764961Google Scholar
  13. R. G. Haydon: A counterexample to several questions about scattered compact spaces, Bull. London Math. Soc. 22 (1990) 261–268.https://doi.org/10.1112/blms/22.3.261Google Scholar
  14. R. G. Haydon: Trees and renorming theory, to appear.Google Scholar
  15. M. G. Crandall, P.-L. Lions: Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983) 1–42.https://doi.org/10.1090/S0002-9947-1983-0690039-8Google Scholar
  16. F. H. Clarke, Y. S. Ledyaev: Mean value inequalities in Hilbert spaces, Trans. Amer. Math. Soc. 344 (1994) 307–324.https://doi.org/10.1090/S0002-9947-1994-1227093-8Google Scholar
  17. M. G. Crandall, P.-L. Lions: Hamilton-Jacobi equations in infinite dimensions, J. Funct. Anal. Part I: Uniqueness of viscosity solutions, 62 (1985) 379–396; Part II: Existence of viscosity solutions, 65 (1986) 368–405; Part III: 68 (1986) 214–247; Part IV: Unbounded linear terms, 90 (1990) 237–283; Part V: B-continuous solutions, 97 (1991) 417–465; Part VI: Nonlinear A and Tataru’s method refined; Part VII: The HJB equation is not always satisfied.Google Scholar
  18. M. G. Crandall, H. Ishii and P.-L. Lions: User’s guide to viscosity solutions of second order fully non linear partial differential equations, Bull. Amer. Math. Soc. 27 (1992) 1–67.https://doi.org/10.1090/S0273-0979-1992-00266-5Google Scholar
  19. A. D. Ioffe, J. P. Penot: On second order subdifferentials, in preparation.Google Scholar
  20. H. Ishii: Perron’s method for Hamilton-Jacobi equations, Duke Math. J. 55 (1987) 369–384.https://doi.org/10.1215/S0012-7094-87-05521-9Google Scholar
  21. G. Barles: Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques et Applications 17, Springer-Verlag, 1994.Google Scholar
  22. L. C. Evans: Some Min-Max Methods for the Hamilton-Jacobi Equation, Indiana Univ. Math. J. 33 (1984) 31–50.https://doi.org/10.1512/iumj.1984.33.33002Google Scholar
  23. H. Frankowska: On the single-valuedness of Hamilton-Jacobi operators, Non Linear Anal. T. M. A. 10 (1986) 1477–1483.https://doi.org/10.1016/0362-546X(86)90117-3Google Scholar

Suggested citation

E. El Haddad, R. Deville. “The Viscosity Subdifferential of the Sum of Two Functions in Banach Spaces. I: First Order Case.” Journal of Convex Analysis 3 (1996), No. 2, 295–308. https://doi.org/10.68381/jca03020

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 1996. Rights now held by Banach Press.