Skip to main content

The Calculus of Variations and Some Semilinear Variational Inequalities of Elliptic and Parabolic Type

  • Chapter
Partial Differential Equations and the Calculus of Variations

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 1))

  • 601 Accesses

Abstract

We present here a survey on some techniques of nonsmooth calculus of variations, that we developed in these last years in collaboration with other authors, and some results concerning semilinear variational inequalities that can be deduced by means of such techniques.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H.Amann, Fixed point equations and nonlinear eigenvalue in ordered Banach spaces, Siam Rew. 18 (1986), 620–709.

    Article  MathSciNet  Google Scholar 

  2. H.Amann, Saddle points and multiple solutions of differential equations, Math.Z. 169 (1979).

    Google Scholar 

  3. H.Amann, E.Zehnder, Non trivial solutions for a class of non resonance problems and applications to nonlinear differential equations, Ann. Sc. Norm. Sup. Pisa 7 (1980), 539–603.

    MathSciNet  MATH  Google Scholar 

  4. A.Ambrosetti, Elliptic equations with jumping nonlinearities, J. Math. and Phys. Sc. 18 (1984).

    Google Scholar 

  5. A.Ambrosetti, G.Prodi, On the inversion of some differentiable mappings with singularities between Banacyh spaces, Ann. Mat. Pura e Appl. 93 (1972), 231–246.

    Article  MathSciNet  MATH  Google Scholar 

  6. A.Ambrosetti, P.H.Rabinowitz, Dual variational methods in critical point theory and applications, J.Funct. Anal. 14 (1973), 349–381.

    Article  MathSciNet  MATH  Google Scholar 

  7. H.Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Pubb-lishing Program), Boston, Mass. — London, 1984.

    MATH  Google Scholar 

  8. C.Baiocchi, A.Capelo, Disequazioni variazionali e quasivaria-zionali, Pitagora, 1978.

    Google Scholar 

  9. R.Böhme, Die Lösung der Verzweigunsgleichungen für nichtlineare Eigenwertprobleme, Math.Z. 127 (1972), 105–126.

    Article  MathSciNet  MATH  Google Scholar 

  10. H.Brezis, Opérateurs maximaux monotone et semigroupes de contraction dans les espaces de Hilbert, North Holland Mathematics Studies 5, Notas de Matematica 50, Amsterdam, London 1973.

    Google Scholar 

  11. H.Brezis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contribution to nonlinear analysis, Zarantonello 1971, 101–156.

    Google Scholar 

  12. H.Brezis, J.M.Coron, E.H.Lieb, Harmonic maps with defects, Comm. Math. Phys. 107 (1986), 649–705.

    Article  MathSciNet  MATH  Google Scholar 

  13. H.Brezis, G.Stampacchia, Sur la regularité de la solution d’in-equations elliptiques, Bull. Soc. Math. France 96 (1968), 153–180.

    MathSciNet  MATH  Google Scholar 

  14. F.E.Brodwer, Non linear elliptic boundary value problems, Bull. Am. Math. Soc. 69 (1963), 862–874.

    Article  Google Scholar 

  15. F.E.Brodwer, Non linear monotone operators and convex subsets in Banach spaces, Bull. Am. Math. Soc. 71 (1965), 780–785.

    Article  Google Scholar 

  16. A.Canino, On p-convex sets and geodesics, J. Diff. Eq., in press.

    Google Scholar 

  17. G.Čobanov, A.Marino, D.Scolozzi, Evolution equations for the eigenvalue problem for the Laplace operator with respect to an obstacle, submitted to Ann. Scuola Normale Sup. Pisa.

    Google Scholar 

  18. G.Čobanov, A.Marino, D.Scolozzi, Multiplicity of eigenvalues of the Laplace operator with respect to an obstacle and non tan-cency conditions, Nonlin. Anal. Th. Meth. Appl, to appear.

    Google Scholar 

  19. G.Čobanov-D.Scolozzi, to appear.

    Google Scholar 

  20. E.De Giorgi, M.Degiovanni, A.Marino, M.Tosques, Evolution equations for a class of nonlinear operators, Atti Accadem. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 75 (1983), 1–8.

    MATH  Google Scholar 

  21. E.De Giorgi, T.Franzoni, Su un tipo di convergenza variazio-nale, Rend. Sem. Mat. Brescia 3 (1979), 63–101.

    Google Scholar 

  22. E.De Giorgi, A.Marino, M.Tosques, Problemi di evoluzione in spazi metrici e curve di massima pendenza, Atti Accadem. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 68 (1980), 180–187.

    MATH  Google Scholar 

  23. E.De Giorgi, A.Marino, M.Tosques, Funzioni (p,q)-convesse, Atti Accadem. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 73 (1982), 6–14.

    MATH  Google Scholar 

  24. M.Degiovanni, Parabolic equations with time-dependent boundary conditions, Ann. Mat. Pura Appl. 141 (1985), 223–263.

    Article  MathSciNet  MATH  Google Scholar 

  25. M.Degiovanni, Bifurcation problems for nonlinear elliptic variational inequalities, Preprint Dip. Mat. Pisa 228 (1988).

    Google Scholar 

  26. M.Degiovanni, On the buckling of a thin elastic plate subjected to unilateral constraints, Nonlinear variational problems (Isola d’Elba, 1986), Proceedings, in press.

    Google Scholar 

  27. M.Degiovanni, Bifurcation for odd nonlinear elliptic variational inequalities, (preprint), Dip. Mat. Pisa 235, Pisa, (1988).

    Google Scholar 

  28. M.Degiovanni, A.Marino, Nonsmooth variational bifurcation, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 81 (1987), 259–269.

    MathSciNet  MATH  Google Scholar 

  29. M.Degiovanni, A.Marino, M.Tosques, General properties of (p,q)-convex functions and (p,q)-monotone operators, Ricerche Mat. 32 (1983), 285–319.

    MathSciNet  MATH  Google Scholar 

  30. M.Degiovanni, A.Marino, M.Tosques, Evolution equations associated with (p,q)-convex functions and (p,q)-monotone operators, Ricerche Mat. 33 (1984), 81–112.

    MathSciNet  MATH  Google Scholar 

  31. M.Degiovanni, A.Marino, M.Tosques, Evolution equations with lack of convexity, Nonlinear Anal. The. Meth. Appl. 12 (1985), 1401–1443.

    Article  MathSciNet  Google Scholar 

  32. M.Degiovanni, M.Tosques, Evolution equations for (ϕ,f)-mo-notone operators, Boll. Un. Mat. It. B 5 (1986), 537–568.

    MathSciNet  MATH  Google Scholar 

  33. C.Do, Bifurcation theory for elastic plates subjected to unilateral conditions, J.Math. Anal. Appl. 60 (1977), 435–448.

    Article  MathSciNet  MATH  Google Scholar 

  34. M.Edelstein, On nearest points of sets in uniformly convex Banach spaces, J. London Math. Soc. 43 (1968), 375–377.

    Article  MathSciNet  MATH  Google Scholar 

  35. G.Fichera, Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno, Mem. Accad. Naz. Lincei 7 (1964), 91–140.

    MathSciNet  MATH  Google Scholar 

  36. M.Frigon, A.Marino, C.Saccon, to appear.

    Google Scholar 

  37. F.Giannoni, to appear.

    Google Scholar 

  38. P.Hartman, G.Stampacchia, On some nonlinear elliptic differential-functional equations, Acta Math. 115 (1966), 271–310.

    Article  MathSciNet  MATH  Google Scholar 

  39. H.Hofer, Variational and tological methods in partially ordered Hilbert spaces, Math. Ann. 261 (1982), 493–514.

    Article  MathSciNet  MATH  Google Scholar 

  40. D.Kinderleherer, G.Stampacchia, An introduction to variational inequalities and their applications, Pure and applied Mathematics, 88, Academic Press, New York, London, Toronto, 1980.

    Google Scholar 

  41. M.A.Krasnoselskii, Topological Methods in the theory of nonlinear integral equations, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956. The Macmillan Co., New York, 1964.

    MATH  Google Scholar 

  42. M.Kŭcera, A new method for obtaining eigenvalues of variational inequalities: operators with multiple eigenvalues, Czechoslovak Math. J. 32 (1982), 197–207.

    MathSciNet  Google Scholar 

  43. M.Kŭcera, Bifurcation point of variational inequalities, Czechoslovak Math. J. 32 (1982), 208–226.

    MathSciNet  Google Scholar 

  44. M.Kŭcera, A global bifurcation theorem for obtaining eigenvalues and bifurcation points, Czechoslovak Math. J. 38 (1988), 120–137.

    MathSciNet  Google Scholar 

  45. H.Lewy, G.Stampacchia, On the regularity of the solutions of a variational inequality, Comm. Pure Appl. Math. 22 (1969), 153–188.

    Article  MathSciNet  MATH  Google Scholar 

  46. J.L.Lions, Quelques méthodes de résolutions des problèmes aux limites non linéaires, Dunod Gauthier-Villars, 1969.

    Google Scholar 

  47. J.L.Lions, G.Stampacchia, Variational inequalities, Comm. Pure and Appl. Math. 20 (1967), 493–519.

    Article  MathSciNet  MATH  Google Scholar 

  48. A.Marino, La biforcazione nel caso variazionale, Confer. Sem. Mat. Univ. Bari 132 (1973).

    Google Scholar 

  49. A.Marino, Evolution equations and multiplicity of critical points with respect to an obstacle, Contributions to modern calculus of variations (Bologna 1985), Res. Notes in Math. 148, Pitman, London — New York, 1987, 123–144.

    Google Scholar 

  50. A.Marino, D.Passaseo, to appear.

    Google Scholar 

  51. A.Marino, C.Saccon, M.Tosques, Curves of maximal slope and parabolic variational inequalities on non convex constraints, Ann. Sc. Norm. Sup. Pisa, to appear.

    Google Scholar 

  52. A.Marino, D.Scolozzi, Punti inferiormente stazionari ed equa-zione di evoluzione con vincoli unilaterali non convessi, Rend. Sem. Mat. e Fis. Milano 52 1982, 393–414.

    Article  MathSciNet  Google Scholar 

  53. A.Marino, D.Scolozzi, Geodetiche con ostacolo, Boll. Un. Mat. It. B 2 (1983), 1–31.

    MathSciNet  Google Scholar 

  54. A.Marino, D.Scolozzi, Autovalori dell’operatore di Laplace ed equazioni di evoluzione in presenza di ostacolo, (Bari 1984) Problemi differenziali e teoria dei punti critici, Pitagora, Bologna, 1984, 137–155.

    Google Scholar 

  55. A.Marino, D.Scolozzi, to appear.

    Google Scholar 

  56. A.Marino, M.Tosques, Curves of maximal slope for a certain class of non regular functions, Boll. Un. Mat. It. B 1 (1982), 143–170.

    MathSciNet  MATH  Google Scholar 

  57. E.Miersemann, Eigenwertaufgaben für Variationsungleichungen, Math. Nachr. 100 (1981), 221–228.

    Article  MathSciNet  MATH  Google Scholar 

  58. E.Miersemann, On higher eigenvalues of variational inequalities, Comment. Math. New Carolin. 24 (1983), 657–665.

    MathSciNet  MATH  Google Scholar 

  59. E.Miersemann, Eigenvalue problems in convex sets, Mathematical Control Theory, 401–408, Banach Center Pubbl. 14, PWN, Warsaw, 1985.

    Google Scholar 

  60. E.Mitidieri, M.Tosques, Volterra integral equations associated with a class of nonlinear operators in Hilbert spaces, Ann. Fac. Sci. Tolouse Math., (5) 8 2, (1986–87).

    MathSciNet  Google Scholar 

  61. E.Mitidieri, Tosques, Nonlinear integrodifferential equations in Hilbert spaces: the variational case, Proceedings of the congress “Volterra integral equations in Banach spaces and applications” (Trento, feb. 1987), to appear.

    Google Scholar 

  62. L.Niremberg, Variational and topological methods in nonlinear problems, Bull. A.M.S. 4 (1981), 267–302.

    Article  Google Scholar 

  63. R.Palais, S.Smale, A generalized Morse theory, Bull. A.M.S. 70 (1964).

    Google Scholar 

  64. D.Passaseo, Molteplicitá di soluzioni per disequazioni variazionali, Tesi di dottorato, Pisa 1988.

    Google Scholar 

  65. D.Passaseo, Molteplicitá di soluzioni per certe disequazioni variazionali di tipo ellittico, Preprint Dip. Mat. Pisa 236, Pisa, 1988.

    Google Scholar 

  66. D.Passaseo, Molteplicitá di soluzioni per disequazioni variazionali non lineari di tipo ellittico, Preprint Dip. Mat. Pisa 248, Pisa, 1988.

    Google Scholar 

  67. P.Quittner, Solvability and multiplicity results for variational inequalities, (preprint), 1987.

    Google Scholar 

  68. P.H.Rabinowitz, Minimax methods in critical point theory with application to differential equations, CBMS Regional Conference Series in Mathematics 65, American Mathematical Society, Providence,R.I., 1986.

    Google Scholar 

  69. R.C.Riddel, Eigenvalue problems for nonlinear elliptic variational inequalities, Nonlin. Anal. Th. Meth. Appl. 3 (1979), 1–33.

    Article  Google Scholar 

  70. C.Saccon, Some parabolic equations on non convex constraints, Boll. Un. Mat. It., to appear.

    Google Scholar 

  71. C.Saccon, On a evolution problem with free boundary, Houston J. Math., to appear.

    Google Scholar 

  72. D.Scolozzi, Esistenza e molteplicitá di geodetiche con ostacolo con estremi variabili, Ricerche Mat. 33 (1984), 171–201.

    MathSciNet  MATH  Google Scholar 

  73. D.Scolozzi, Un teorema di esistenza di una geodetica chiusa su varietá con bordo, Boll. Un. Mat. It. A 4 (1985), 451–457.

    MathSciNet  Google Scholar 

  74. M.Tosques, Quasi autonomous evolution equations associated with (ϕ, f)-monotone operators, (Trieste 1985), Rend. Circ. Mat. Palermo Suppl. 15 (1987), 163–180.

    MathSciNet  Google Scholar 

  75. G.Stampacchia, Formes bilinépaires coercitives sur les ensembles convexes, C. R. Acad. Sc. Paris 258 (1964), 4413–4416.

    MathSciNet  MATH  Google Scholar 

  76. G.Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre a coefficents discontinus, Ann. Inst. Fourier Grenoble 15 (1965), 189–257.

    Article  MathSciNet  MATH  Google Scholar 

  77. G.Stampacchia, Regularity of solutions of some variational inequalities, Proceeding of Symposia on Nonlinear Functional Analysis 18 I (1980).

    Google Scholar 

  78. A.Szulkin, On a class of variational inequalities involving gradient operators, J. Math. Annal. Appl. 100 (1984).

    Google Scholar 

  79. A.Szulkin, On the solvability of a class of semilinear variational inequalities, Rend. Mat. 4 (1984).

    Google Scholar 

  80. A.Szulkin, Positive solutions of variational inequalities: a degree theoretical approach, J. Diff. Eq. 57 (1985), 90–111.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Additional information

Dedicated to Ennio De Giorgi on his sixtieth birthday

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer Science+Business Media New York

About this chapter

Cite this chapter

Marino, A. (1989). The Calculus of Variations and Some Semilinear Variational Inequalities of Elliptic and Parabolic Type. In: Colombini, F., Marino, A., Modica, L., Spagnolo, S. (eds) Partial Differential Equations and the Calculus of Variations. Progress in Nonlinear Differential Equations and Their Applications, vol 1. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4615-9831-2_12

Download citation

  • DOI: https://doi.org/10.1007/978-1-4615-9831-2_12

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4615-9833-6

  • Online ISBN: 978-1-4615-9831-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics