Buckling of nonlinearly elastic rods in the presence of obstacles treated by nonsmooth critical point theory

Mathematische Annalen

Volumn 311, Issue 4, 1998, Pages 675-728

  Degiovanni, Marco ^a   Schuricht, Friedemann ^b  

^a UNIVERSITÀ CATTOLICA DEL SACRO CUORE   (Italy)

^b MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032354984     PISSN: 00255831     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002080050206     Document Type: Article

References (50)

1. Ambrosetti, A.: Variational methods and nonlinear problems: classical results and recent advances. In: Matzeu, M., Vignoli, A. (eds.) Topological Nonlinear Analysis. Degree, singularity, and Variations, (Progress in Nonlinear Differential Equations and their Applications, vol. 15), pp 1-36, Boston: Birkhäuser 1995
2. Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349-381 (1973)
3. Antman, S.S.: The theory of rods. In: Truesdell, C. (ed.) Handbuch der Physik, vol. VIa/2, pp 641-703, Berlin: Springer 1972
4. Antman, S.S.: Qualitative theory of ordinary differential equations of nonlinear elasticity. In: Nemat-Nasser, S. (ed.) Mechanics Today, vol. 1, pp 58-101, New York: Pergamon Press 1972
5. Antman, S.S.: The influence of elasticity on analysis: modern developments. Bull. Amer. Math. Soc. (New Series) 9, 267-291 (1983)
6. Antman, S.S.: Large lateral buckling of nonlinearly elastic beams. Arch. Rational Mech. Anal. 84, 293-305 (1983)
7. Antman, S.S.: Nonlinear Problems of Elasticity, New York: Springer 1995
8. Antman, S.S., Marlow, R.S.: Transcritical buckling of columns. Z. angew. Math. Phys. 43, 7-27 (1992)
9. Antman, S.S., Marlow, R.S.: New phenomena in the buckling of arches described by refined theories. Int. J. Solids Structures 30,16, 2213-2241 (1993)
10. Antman, S.S. Pierce, J.F.: The intricate global structure of buckled states of compressible columns. SIAM J. Appl. Math. 50, 395-419 (1990)
11. Antman, S.S., Rosenfeld, G.: Global behavior of buckled states of nonlinearly elastic rods. SIAM Review 20, 513-566 (1978), Corrections and additions: ibid. 22, 186-187 (1980)
12. Aubin, J.-P., Ekeland, I.: Applied Nonlinear Analysis, Pure and Applied Mathematics. New York: John Wiley & Sons 1984
13. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Boston: Birkhäuser 1990
14. Barbu, V., Precupanu, T.: Convexity and Optimization in Banach spaces. Bucaresti: Ed. Acad.; Dordrecht: D. Reidel Publ. Co. 1986
15. Canino, A., Degiovanni, M.: Nonsmooth critical point theory and quasilinear elliptic equations. In: Topological Methods in Differential Equations and Inclusions (Montreal, 1994), pp 1-50, NATO ASI Series C 472, Dordrecht: Kluwer Academic Publishers 1995
16. Canino, A., Perri, U.: Constrained problems in Banach spaces with an application to variational inequalities. Nonlinear Anal. 24, 839-856 (1995)
17. Chang, K.C.: Variational methods for non-differentiable functional and their applications to partial differential equations. J. Math. Anal. Appl. 80, 102-129 (1981)
18. Clark, D.C.: A variant of the Ljusternik-Schnirelmann theory. Indiana Univ. Math. J. 22, 65-74 (1972)
19. Clarke, F.H.: Optimization and Nonsmooth Analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts, New York: John Wiley & Sons 1983
20. Corvellec, J.-N.: Morse theory for continuous functionals. J. Math. Anal. Appl. 196, 1050-1072 (1995)
21. Corvellec, J.N.: A note on coercivity of lower semicontinuous functions and nonsmooth critical point theory. Serdica Math. J., 22, 57-68 (1996)
22. Corvellec, J.-N., Degiovanni, M., Marzocchi, M.: Deformation properties for continuous functionals and critical point theory. Topol. Methods Nonlinear Anal. 1, 151-171 (1993)
23. Courant, R.: Über die Eigenwerte bei den Differentialgleichungen der Mathematischen Physik. Math. Z. 7, 1-57 (1920)
24. Degiovanni, M., Marzocchi, M.: A critical point theory for nonsmooth functionals. Ann. Mat. Pura Appl. (4) 167, 73-100 (1994)
25. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Boca Raton: CRC Press 1992
26. Fischer, E.: Über quadratische Formen mit reellen Koeffizienten. Monatsh. Math. Phys. 16, 234-249 (1905)
27. Ioffe, A., Schwartzman, E.: Metric critical point theory 1. Morse regularity and homotopic stability of a minimum. J. Math. Pures Appl. 75, 125-153 (1996)
28. Katriel, G.: Mountain pass theorems and global homeomorphism theorems. Ann. Inst. H. Poincaré Anal. Non Linéaire 11, 189-209 (1994)
29. Ljusternik, L.: Topologische Grundlagen der allgemeinen Eigenwerttheorie. Monatsh. Math. Phys. 37, 125-130 (1930)
30. Maddocks, J.H.: Stability of nonlinearly elastic rods. Arch. Rational Mech. Anal. 85, 311-354 (1984)
31. Matzeu, M., Vignoli, A. (eds): Topological Nonlinear Analysis. Degree, Singularity and Variations. (Progress in Nonlinear Differential Equations and their Applications, vol. 15) Boston: Birkhäuser 1995
32. Miersemann, E.: Über höhere Verzweigungspunkte nichtlinearer Variationsungleichungen. Math. Nachr. 85, 195-213 (1978)
33. Miersemann, E.: Eigenvalue problems for variational inequalities. Contemp. Math. 4, 25-43 (1981)
34. Morse, M.: Relations between the critical points of a real function of n independent variables. Trans. Amer. Math. Soc. 27, 345-396 (1925)
35. Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. New York: M. Dekker 1995
36. Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Boston: Birkhäuser 1985
37. Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Series Math. 65, Amer. Math. Soc., Providence, R.I., 1986
38. Ribarska, N.K., Tsachev, Ts.Y., Krastanov, M.I.: Speculating about mountains, Well-posed Problems and Stability in Optimization (Marseille, 1995), Serdica Math. J. 22, 341-358 (1996)
39. Rockafellar, R.T.: Integral functionals, normal integrands and measurable selections. In: Gossez, J.G., Lami Dozo, E.J., Mawhin, J., Waelbroeck, L. (eds.) Nonlinear Operators and the Calculus of Variations (Lecture Notes in Mathematics, vol. 543) pp 157-207, Berlin: Springer 1976
40. Schnirelman, L.: Über eine neue kombinatorische Invariante. Monatsh. Math. Phys. 37, 131-134 (1930)
41. Schuricht, F.: Minimax principle for eigenvalue variational inequalities in the nonsmooth case. Math. Nachr. 152, 121-143 (1991)
42. Schuricht, F.: Minimax principle for eigenvalue problems of variational inequalities in convex sets. Math. Nachr. 163, 117-132 (1993)
43. Schuricht, F.: Bifurcation from minimax solutions by variational inequalities in convex sets. Nonlinear Anal. 26, 91-112 (1996)
44. Schuricht, F.: A variational approach to obstacle problems for shearable nonlinearly elastic rods. Arch. Rational Mech. Anal. 140, 103-159 (1997)
45. F. Schuricht, F.: Regularity for shearable nonlinearly elastic rods in obstacle problems, to appear in Arch. Rational Mech. Anal.
46. Struwe, M.: Variational Methods. Berlin: Springer 1990
47. Szulkin, A.: Minimax principles for lower semicontinuous functions and application to nonlinear boundary value problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 3, 77-109 (1986)
48. Weyl, H.: Über die asymptotische Verteilung der Eigenwerte. Göttinger Nachrichten, 110-117 (1911)
49. Zeidler, E.: Nonlinear Functional Analysis and its Applications, Vol. I: Fixed-Point Theorems. New York: Springer 1986
50. Zeidler, E.: Nonlinear Functional Analysis and its Applications, Vol. IIB: Nonlinear Monotone operators. New York: Springer 1990