The search session has expired. Please query the service again.

Eigenvalue problems for some variational inequalities with pointwise gradient constraint

Claudio Saccon

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993)

  • Volume: 4, Issue: 3, page 185-195
  • ISSN: 1120-6330

Abstract

top
Some eigenvalue problems for elliptic semilinear variational inequalities are studied, the main feature being the presence of an obstacle on the first derivative of the unknown function. The role of a «nontangency» assumption is put into evidence: to have existence and multiplicity results one has to check that the convex set, produced by the obstacle condition, and the sphere in the function space, on which it seems natural to study eigenvalue problems, are not tangent. This condition is studied in some problems of the fourth and of the second order and some sufficient conditions for it are found, which allow to get results of existence and multiplicity.

How to cite

top

Saccon, Claudio. "Autovalori di alcune disequazioni variazionali con vincoli puntati sulle derivate." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 4.3 (1993): 185-195. <http://eudml.org/doc/244268>.

@article{Saccon1993,
abstract = {Si studiano problemi di autovalori per disequazioni variazionali semilineari ellittiche con un ostacolo puntuale sulla derivata prima della funzione incognita. Si mette in particolare in evidenza il ruolo della «ipotesi di non tangenza» tra il convesso, che viene definito dalla condizione di ostacolo, e la sfera dello spazio funzionale, su cui è naturale studiare un problema di autovalori. Tale condizione viene analizzata in alcuni casi concreti e si indicano alcune ipotesi che, garantendone la validità, danno luogo ad alcuni risultati di esistenza e molteplicità.},
author = {Saccon, Claudio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Eigenvalue problems; Variational inequalities; Nonsmooth analysis; nontangency; elliptic semilinear variational inequalities},
language = {ita},
month = {9},
number = {3},
pages = {185-195},
publisher = {Accademia Nazionale dei Lincei},
title = {Autovalori di alcune disequazioni variazionali con vincoli puntati sulle derivate},
url = {http://eudml.org/doc/244268},
volume = {4},
year = {1993},
}

TY - JOUR
AU - Saccon, Claudio
TI - Autovalori di alcune disequazioni variazionali con vincoli puntati sulle derivate
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1993/9//
PB - Accademia Nazionale dei Lincei
VL - 4
IS - 3
SP - 185
EP - 195
AB - Si studiano problemi di autovalori per disequazioni variazionali semilineari ellittiche con un ostacolo puntuale sulla derivata prima della funzione incognita. Si mette in particolare in evidenza il ruolo della «ipotesi di non tangenza» tra il convesso, che viene definito dalla condizione di ostacolo, e la sfera dello spazio funzionale, su cui è naturale studiare un problema di autovalori. Tale condizione viene analizzata in alcuni casi concreti e si indicano alcune ipotesi che, garantendone la validità, danno luogo ad alcuni risultati di esistenza e molteplicità.
LA - ita
KW - Eigenvalue problems; Variational inequalities; Nonsmooth analysis; nontangency; elliptic semilinear variational inequalities
UR - http://eudml.org/doc/244268
ER -

References

top
  1. BERGER, M. S., Nonlinearity and Funtional Analysis. Academic Press, New York-San Francisco-London1977. Zbl0368.47001MR488101
  2. COBANOV, G. - MARINO, A. - SCOLOZZI, D., Multiplicity of eigenvalues of the Laplace operator with respect to an obstacle and non tangency conditions. Nonlinear Anal. Th. Meth. Appl., vol. 15, 3, 1990, 199-215. Zbl0716.49009MR1065252DOI10.1016/0362-546X(90)90157-C
  3. COBANOV, G. - MARINO, A. - SCOLOZZI, D., Evolution equations for the eigenvalue problem for the Laplace operator with respect to an obstacle. Rend. Accad. Naz. Sci. XL, Mem. Mat., 14, 1990, 139-162. Zbl0729.35088MR1106575
  4. DE GIORGI, E. - MARINO, A. - TOSQUES, M., Problemi di evoluzione in spazi metrici e curve di massima pendenza. Atti Acc. Lincei Rend. fis., s. 8, vol. 68, 1980, 180-187. Zbl0465.47041MR636814
  5. DEGIOVANNI, M., Bifurcation problems for nonlinear elliptic variational inequalities. Ann. Fac. Sci. Toulouse, 10, 1989, 215-258. Zbl0656.58030MR1425487
  6. DEGIOVANNI, M., Homotopical properties of a class of nonsmooth functions. Ann. Mat. Pura e Applicata, vol. CLVI, 1990, 37-71. Zbl0722.58013MR1080210DOI10.1007/BF01766973
  7. DEGIOVANNI, M. - MARINO, A., Nonsmooth variational bifurcation. Atti Acc. Lincei Rend. fis., s. 8, vol. 81, 1987, 259-269. Zbl0671.58029MR999818
  8. DEGIOVANNI, M. - MARINO, A. - TOSQUES, M., Evolution equations with lack of convexity. Nonlinear Anal. Th. Meth. Appl., (9), 12, 1985, 1401-1443. Zbl0545.46029MR820649DOI10.1016/0362-546X(85)90098-7
  9. DO, C., Bifurcation theory for elastic plates subjected to unilateral conditions. J. Math. Anal. Appl., 60, 1977, 435-448. Zbl0364.73030MR455672
  10. KRASNOSELSKII, M. A., Topological Methods in the Theory of Nonlinear Integral Equations. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow1956. The Macmillan Co., New York1964. MR159197
  11. KUČERA, M., A new method for obtaining eigenvalues of variational inequalities: operators with multiple eigenvalues. Czechoslovak Math. J., 32, 1982, 197-207. Zbl0621.49005MR654056
  12. KUČERA, M., Bifurcation point of variational inequalities. Czechoslovak Math. J., 32, 1982, 208-226. Zbl0621.49006MR654057
  13. KUČERA, M., A global bifurcation theorem for obtaining eigenvalues and bifurcation points. Czechoslovak Math. J., 38, 1988, 120-137. Zbl0665.35010MR925946
  14. KUČERA, M. - NEČAS, J. - SOUČEK, J., The eigenvalue problem for variational inequalities and a new version of the Lustemik-Schnirelmann theory. In: Nonlinear Analysis. Collection of papers in honour Erich H. Rothe. Academic Press, New York1978, 125-143. Zbl0463.47041MR513782
  15. MARINO, A. - SACCON, C. - TOSQUES, M., Curves of maximal slope and parabolic variational inequalities on non convex constraints. Annali Sc. Norm. Sup. Pisa, vol. XVI, 1989, 281- 330. Zbl0699.49015MR1041899
  16. MARINO, A. - SCOLOZZI, D., Geodetiche con ostacolo. Boll. Un. Mat. Ital., B (6) 2, 1983, 1-31. MR698480
  17. MARINO, A. - TOSQUES, M., Some variational problems with lack of convexity and some partial differential inequalities. In: Methods of Nonconvex Analysis. Lecture notes in math., 1446, Springer-Verlag, 1989, 58-83. Zbl0716.49010MR1079759DOI10.1007/BFb0084931
  18. MIERSEMANN, E., Eigenwertaufgaben für Variationsungleichungen. Math. Nachr., 100, 1981, 221-228. Zbl0474.49011MR632628DOI10.1002/mana.19811000112
  19. MIERSEMANN, E., On higher eigenvalues of variational inequalities. Comment. Math. Univ. Carolin., 24, 1983, 657-665. Zbl0638.49020MR738561
  20. MIERSEMANN, E., Eigenvalue problems in convex sets. Mathematical Control Theory: 401-408, Banach Center Pubbl., 14PWN, Warsaw1985. MR851239
  21. QUITTNER, P., Spectral analysis of variational inequalities. Comment. Math. Univ. Carolin., 27, 1986, 605-629. Zbl0652.49008MR873631
  22. RABINOWITZ, P. H., Variational methods for nonlinear eigenvalue problems. C.I.M.E., Varenna 1974, 1-56. Zbl0278.35040MR464299
  23. RIDDEL, R. C., Eigenvalue problems for nonlinear elliptic variational inequalities. Nonlinear Anal. Th. Meth. Appl., 31979, 1-33. Zbl0416.49009
  24. SACCON, C., Some parabolic equations on nonconvex constraints. Boll. Un. Mat. Ital., B (7) 3, 1989, 369-385. Zbl0716.35035MR998002
  25. SACCON, C., On the eigenvalues of a fourth order elliptic variational inequality with pointwise gradient constraint. Preprint Dip. Mat. Pisa, ottobre 1992. 
  26. SCHURICHT, F., Minimax principle for eigenvalue variational inequalities in the nonsmooth case. Math. Nachr., 152, 1991, 121-143. Zbl0745.49012MR1121229DOI10.1002/mana.19911520112
  27. SCHURICHT, F., Bifurcation from minimax solutions by variational inequalities. Math. Nachr., 154, 1991, 67-88. Zbl0762.49019MR1138371DOI10.1002/mana.19911540107
  28. SZULKIN, A., On a class of variational inequalities involving gradient operators. J. Math. Annal. Appl., 100, 1984, 486-499. Zbl0551.49008MR743337DOI10.1016/0022-247X(84)90097-0
  29. SZULKIN, A., On the solvability of a class of semilinear variational inequalities. Rend. Mat., (7) 4, 1984, 121-137. Zbl0608.35003MR807126
  30. SZULKIN, A., Posive solutions of variational inequalities: a degree theoretical approach. J. Differential Equations, 57, 1985, 90-111. Zbl0535.35029MR788424DOI10.1016/0022-0396(85)90072-5

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.