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NSA-UPRH Regional Conference on Global Continuation Methods

Project Description

Historically the problems posed by continuum mechanics have induced major developments in important areas of mathematics, such as the theory of partial differential equations both linear and nonlinear, the calculus of variations, and bifurcation theory (cf. [15], [2]). On the other hand, there has been an increasing interest among the mathematical community in nonlinear problems and techniques to tackle such problems. An example of this has been the use of degree theoretic techniques, e.g. the Leray-Schauder degree in nonlinear differential equations. These mutual influences between continuum mechanics and several areas of mathematics research are still going on. In this conference series we will study some important recent developments, explained in more details below, in degree theoretic techniques and their applications to problems in nonlinear elastostatics. These recent developments in this field have an applicability that extends beyond its original motivation from elasticity theory and consequently are worth knowing by other mathematicians interested in nonlinear problems and theories. Moreover, this area of research has many interesting open problems and would benefit from new researchers, specially young talented people.

For many years one of the most difficult open problems of non-linear elasticity theory has been the use of global continuation methods (via degree theory) to study the governing system of partial differential equations of three-dimensional models, c.f. [4], and [13]. The use of Leray-Schauder degree techniques in elasticity has a long and successful story that we will not review here but we refer to [3] for examples and its extensive literature review. However, for the most part, those applications have been limited to one-dimensional problems. Not until recently, in [6], such a major enterprise was carried out for the three dimensional displacement problem of nonlinear elasticity. On the other hand, the full nonlinearity of traction boundary conditions renders more general boundary value problems out of reach to the traditional Leray-Schauder degree.

A more general degree based on proper Fredholm maps of index zero ([5], [12], [11]) avoids the transformation of the original problem into one in terms of a compact perturbation of the identity but requires properness of the nonlinear operator and some a priori estimates on solutions of the linear problem and its spectrum. This generalized degree has the same important properties of the classical Leray-Schauder degree. In particular, the homotopy invariance property (cf. Proposition 4.12 in [11]) of the new degree supports its applicability to study global bifurcation in the sense of Rabinowitz (c.f. [14]). For the three dimensional mixed problem of nonlinear elasticity the required spectral estimates were obtained in [11] and together with the estimates of [1] for elliptic systems, Healey and Simpson were able to apply the generalized degree to get the existence of a global branch of solutions of this problem. Therefore, the new methods of Healey and Simpson make it possible, for the first time, to tackle global bifurcation problems in non-linear three-dimensional elasticity.

A first step in the analysis of local continuation or bifurcation is to study each linearized problem around the corresponding trivial solution. For global continuation or bifurcation one further needs to study the linearized problem about an arbitrary deformation. These linear problems correspond to elliptic systems of partial differential equations on a domain determined by the geometry of the physical problem. In order to apply the more general degree mentioned above, the linear operators must be Fredholm of index zero and certain spectral estimates are needed. When the domain is smooth, one can use Schauder estimates ([1]) to get the required properties. On other types of regions one might use hidden symmetries in the problem to get the required properties ([7], [8], [9]).

The behavior of the global solution branches predicted in [11] is characterized, in addition to the two Rabinowitz alternatives, cf. [14], by the possibility that they terminate due to loss of local injectivity and/or ellipticity and/or the failure of the complementing condition. An open problem then is to find physically meaningful restrictions on the constitutive laws which rule out some of these alternatives. In [10] the failure of local injectivity on bounded solution branches is obviated for a general class of stored energy functions subject to mild growth conditions. Thus, the existence of unbounded solution branches is obtained. With slightly stronger, but nonetheless, physically realistic, growth conditions, a similar result is obtained in [9] for a class of boundary value problems involving traction-free boundary conditions.

References

  1. S. Agmon, A. Douglis, and L. Nirenberg. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Comm. Pure Appl. Math., II(17):35-92, 1964.

  2. S. S. Antman. The influence of elasticity on analysis: Modern developments. Bull. Amer. Math. Soc., 9(3):267-291, 1983.

  3. S. S. Antman. Nonlinear Problems of Elasticity, volume 107 of Applied Mathematical Sciences. Springer Verlag, New York, 1995.

  4. S. S. Antman. Continuation methods in nonlinear elasticity. In V. Lakshmikantham, editor, Proceedings of 1st World Congress of Nonlinear Analysis, 1992, pages 827-838. De Gruyter, 1996.

  5. C. C. Fenske. Extensio gradus ad quasdam applicationes fredholmii. Mitt. Math. Seminar, Giessen, pages 65-70, 1976.

  6. T. J. Healey. Global continuation in displacement problems of nonlinear elastostatics via the leray-schauder degree. Arch. Rational Mech. Anal., pages 273-282, 2000.

  7. T. J. Healey and H. Kielhöfer. Symmetry and nodal properties in the global bifurcation analysis of quasi-linear elliptic equations. Arch. Rat. Mech. Anal., 113:299-311, 1991.

  8. T. J. Healey and H. Kielhöfer. Symmetry and preservation of nodal structure in elliptic equations satisfying fully nonlinear newmann boundary conditions. In E.~L. Allgower, K.~Boehmer, and M.~Golubitsky, editors, Bifurcation and Symmetry, volume 104, pages 169-177. Birkhöuser, Basel, 1992.

  9. T. J. Healey and E.L. Montes-Pizarro. Global bifurcation in nonlinear elasticity with an application to barreling of cylinders. Manuscript, 2002.

  10. T. J. Healey and P. Rosakis. Unbounded branches of classical injective solutions to the forced displacement problem in nonlinear elastostatics. Journal of Elasticity, 49:65-78, 1997.

  11. T. J. Healey and H. C. Simpson. Global continuation in nonlinear elasticity. Arch. Rat. Mech. Anal., 143:1-28, 1998.

  12. H. Kielhöfer. Multiple eigenvalue bifurcation for fredholm mappings. J. Reine Angew. Math., pages 104-124, 1985.

  13. J. Marsden and Hughes. Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs, 1983.

  14. P. Rabinowitz. Some global results for nonlinear eigenvalue problems. J. Funct. Anal., 7:487-513, 1971.

  15. C.A. Truesdell. The influence of elasticity on analysis: The classic heritage. Bull. Amer. Math. Soc., 9(3):293-310, 1983.