Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Folds and cusps in Banach spaces with applications to nonlinear partial differential equations. II


Authors: M. S. Berger, P. T. Church and J. G. Timourian
Journal: Trans. Amer. Math. Soc. 307 (1988), 225-244
MSC: Primary 35J65; Secondary 47H15, 58C27
DOI: https://doi.org/10.1090/S0002-9947-1988-0936814-8
MathSciNet review: 936814
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Earlier the authors have given abstract properties characterizing the fold and cusp maps on Banach spaces, and these results are applied here to the study of specific nonlinear elliptic boundary value problems. Functional analysis methods are used, specifically, weak solutions in Sobolev spaces. One problem studied is the inhomogeneous nonlinear Dirichlet problem

$\displaystyle \Delta u + \lambda u - {u^3} = g\quad {\text{on}}\;\Omega ,\qquad u\vert\partial \Omega = 0,$

where $ \Omega \subset {{\mathbf{R}}^n}(n \leqslant 4)$ is a bounded domain. Another is a nonlinear elliptic system, the von Kármán equations for the buckling of a thin planar elastic plate when compressive forces are applied to its edge.

References [Enhancements On Off] (What's this?)

  • [A] R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
  • [AM] A. Ambrosetti and G. Mancini, Sharp nonuniqueness results for some nonlinear problems, Nonlinear Anal. 3 (1979), 635-645. MR 541874 (80k:47073)
  • [An] S. S. Antman, The influence of elasticity on analysis: Modern developments, Bull. Amer. Math. Soc. (N.S.) 9 (1983), 267-291. MR 714990 (85f:01005)
  • [B-1] M. S. Berger, Nonlinearity and functional analysis, Academic Press, New York, 1977. MR 0488101 (58:7671)
  • [B-2] -, Nonlinear problems with exactly three solutions, Indiana Univ. Math. J. 28 (1979), 689-698. MR 542952 (80h:47068)
  • [B-3] -, New applications of the calculus of variations in the large to nonlinear elasticity, Comm. Math. Phys. 35 (1974), 141-150. MR 0394751 (52:15550)
  • [B-4] -, On von Kármán's Equations and the buckling of a thin elastic plate. I; The clamped plate, Comm. Pure Appl. Math. 20 (1967), 687-719. MR 0221808 (36:4860)
  • [BC] M. S. Berger and P. T. Church, Complete integrability and perturbation of a nonlinear Dirichlet problem. I, Indiana Univ. Math. J. 28 (1979), 935-952. Erratum, ibid. 30 (1981), 799. MR 551157 (80k:58097)
  • [BCT-1] M. S. Berger, P. T. Church and J. G. Timourian, An application of singularity theory to nonlinear elliptic partial differential equations, Singularities (P. Orlik, ed.), Proc. Sympos. Pure Math., vol. 40, Part 1, Amer. Math. Soc., Providence, R.I., 1983, pp. 119-126. MR 713051 (84k:58031)
  • [BCT-2] -, Folds and cusps in Banach spaces, with applications to nonlinear partial differential equations. I, Indiana Univ. Math. J. 34 (1985), 1-19. MR 773391 (86i:58017)
  • [BCT-3] -, Integrability of nonlinear differential equations via functional analysis, Nonlinear Functional Analysis and Its Applications (F. E. Browder, ed.), Proc. Sympos. Pure Math., vol. 45, Part 1, Amer. Math. Soc., Providence, R. I., 1986, pp. 117-123. MR 843553 (87k:58031)
  • [BF-1] M. S. Berger and P. C. Fife, On von Kármán's equations and the buckling of a thin elastic plate, Bull. Amer. Math. Soc. 72 (1966), 1006-1011. MR 0203219 (34:3072)
  • [BF-2] -, On von Kármán's equations and the buckling of a thin elastic plate. II: Plate with general boundary conditions, Comm. Pure Appl. Math. 12 (1968), 227-247.
  • [BJS] L. Bers, F. John, and M. Schecter, Partial differential equations, Wiley, New York, 1964. MR 0162045 (28:5247)
  • [CaC] V. Cafagna and P. T. Church, in preparation.
  • [CaD-1] V. Cafagna and F. Donato, Un résult global de multiplicité pour un problème différentiel nonlinéaire du premier ordre, C. R. Acad. Sci. Paris 300 (1985), 523-526.
  • [CaD-2] -, Singularity theory and the number of solutions to some nonlinear differential problems, preprint.
  • [CHM] S. Chow, J. K. Hale and J. Mallet-Paret, Applications of generic bifurcation. I, Arch. Rational Mech. Anal. 59 (1975), 159-188. MR 0390852 (52:11675)
  • [CDT] P. T. Church, E. N. Dancer, and J. G. Timourian, in preparation.
  • [CT] P. T. Church and J. G. Timourian, The singular set of a nonlinear elliptic operator, Michigan Math. J. (to appear). MR 959267 (89i:58020)
  • [CR] P. G. Ciarlet and P. Rabier, Les equations de von Kármán, Lecture Notes in Math., vol. 826, Springer-Verlag, New York, 1980. MR 595326 (82g:73030)
  • [De] K. Deimling, Nonlinear functional analysis, Springer-Verlag, New York, 1985. MR 787404 (86j:47001)
  • [Di] J. Dieudonné, Foundations of modern analysis, Academic Press, New York, 1960. MR 0120319 (22:11074)
  • [GT] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer-Verlag, Berlin, 1983. MR 737190 (86c:35035)
  • [GG] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, New York, 1973. MR 0341518 (49:6269)
  • [HW] W. Hurewitz and H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, N. J., 1941. MR 0006493 (3:312b)
  • [K] V. L. Klee, Jr., A note on topological properties of normed linear spaces, Proc. Amer. Math. Soc. 7 (1956), 673-674. MR 0078661 (17:1227c)
  • [LM] F. Lazzeri and A. M. Micheletti, An application of singularity theory to nonlinear differentiable mappings between Banach spaces, preprint.
  • [M] W. S. Massey, Algebraic topology: An introduction, Springer-Verlag, New York, 1967; fourth corrected printing, 1977. MR 0448331 (56:6638)
  • [Mc] H. P. McKean, Singularities of a simple elliptic operator, preprint. MR 880181 (88d:58119)
  • [McS] H. P. McKean and J. C. Scovel, Geometry of some nonlinear operators, Ann. Pisa 12 (1986), 299-346. MR 876127 (88e:58103)
  • [P] R. Palais, Natural operations on differential forms, Trans. Amer. Math. Soc. 92 (1959), 125-141. MR 0116352 (22:7140)
  • [R-1] B. Ruf, Multiplicity results for nonlinear elliptic equations, preprint. MR 921246 (88k:35071)
  • [R-2] -, Singularity theory and the geometry of a nonlinear elliptic equation, preprint.
  • [Sc] J. T. Schwartz, Nonlinear functional analysis, Gordon and Breach, New York, 1969. MR 0433481 (55:6457)
  • [St] J. J. Stoker, Nonlinear elasticity, Gordon and Breach, New York, 1968. MR 0413654 (54:1768)
  • [Sz] A. Szulkin, On the number of solutions of some semilinear elliptic boundary value problems, Nonlinear Anal. 6 (1982), 95-116. MR 647589 (83d:35054)
  • [Z] E. Zeidler, Nonlinear functional analysis. I. Fixed-point theorems, Springer-Verlag, New York, 1986. MR 816732 (87f:47083)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35J65, 47H15, 58C27

Retrieve articles in all journals with MSC: 35J65, 47H15, 58C27


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0936814-8
Keywords: Nonlinear partial differential equations or systems, elliptic boundary value problem, nonlinear Dirichlet problem, fold map, cusp map, von Kármán equations, bifurcation, singularity theory in infinite dimensions
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society