Abstract
The concept of normal hyperbolicity of center manifolds is generalized to infinite-dimensional differential equations, in particular, to elliptic problems in cylindrical domains. It is shown that all solutions u staying close to the center manifold for t ∈ (−l,l) satisfy an estimate of the form \(\left\| {u(t) - \tilde u(t)} \right\| \leqslant Ce^{ - \alpha (l - |t|)} \) where C and α are independent of l, and ũ is a solution on the center manifold. These results are applied to Saint-Venant's principle for the static deformation of nonlinearly elastic prismatic bodies. The use of the center manifold permits the effective treatment of the general case of non-zero resultant forces and moments acting on each cross-section.
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References
Antman, S. S., & C. S. Kenney [1981]: Large buckled states of nonlinearly elastic rods under torson, thrust, and gravity. Arch. Rational Mech. Anal. 76, 289–338.
Bonic, R., & J. Frampton [1966]: Smooth functions on Banach manifolds, J. Math. Mech. 16, 877–898.
Breuer, S., & J. J. Roseman [1977]: On SV's principle in 3-dimensional nonlinear elasticity, Arch. Rational Mech. Anal. 63, 187–241.
Chow, S.-N., & J. K. Hale [1982]: Methods of Bifurcation Theory, Berlin-Heidelberg-New York: Springer-Verlag.
Ericksen, J. L. [1983]: Problems for infinite elastic prisms—Saint Venant's problem for elastic prisms, in “Systems of Nonlinear Partial Differential Equations”, J. M. Ball ed., NATO ASI C 111, Boston: Reidel Publ. Comp.
Fischer, G. [1984]: Zentrumsmannigfaltigkeiten bei elliptischen Differentialgleichungen, Math. Nachr. 115, 137–157.
Hale, J. K., & J. Scheurle [1985]: Smoothness of bounded solutions of nonlinear evolution equations, J. Diff. Equations 56, 142–163.
Hirsch, M. W., Pugh, C. C., & M. Shub [1977]: Invariant manifolds, Lecture Notes in Mathematics Vol. 583, Berlin-Heidelberg-New York: Springer-Verlag.
Holmes, P. J., & A. Mielke [1988]: Spatially complex equilibria of buckled rods, Arch. Rational Mech. Anal. 101, 319–348.
Horgan, C. O., & J. K. Knowles [1983]: Recent developments concerning Saint-Venant's principle, Advances in Applied Mechanics 23, 179–269.
Kirchgässner, K. [1982]: Wave solutions of reversible systems and applications, J. Diff. Equations 45, 113–127.
Kirchgässner, K., & J. Scheurle [1986]: Saint-Venant's principle from the dynamical point of view, manuscript, Universität Stuttgart.
Kirchhoff, G. [1859]: Über das Gleichgewicht und die Bewegungen eines unendlich dünnen Stabes, Journal für Mathematik (Crelle) 56, 285–313.
Knops, R. J., & L. E. Payne [1983]: A Saint-Venant's principle for nonlinear elasticity, Arch. Rational Mech. Anal. 81, 1–12.
Mielke, A. [1986]: A reduction principle for nonautonomous systems in infinite-dimensional spaces, J. Diff. Equations 65, 68–88.
Mielke, A. [1988a]: Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Meth. Appl. Sciences 10, 51–66.
Mielke, A. [1988b]: Saint-Venant's problem and semi-inverse solutions in nonlinear elasticity. Arch, Rational Mech. Anal. 102, 205–229.
Mielke, A. [1989]: On nonlinear problems of mixed type: a qualitative theory using infinite-dimensional center manifolds, Dynamics, Diff. Eqns., submitted.
Oleinik, O.A., & G. A. Yosifian [1982]: On the asymptotic behaviour at infinity of solutions in linear elasticity, Arch. Rational Mech. Anal. 78, 29–53.
Orazov, B. B. [1989]: On the asymptotic behaviour at infinity of solutions of the traction boundary value problem, Proc. Royal Soc. Edinburgh 111 A, 33–52.
Palmer, K. J. [1975]: Linearization near integral manifolds, J. Math. Anal. Appl. 51, 243–255.
Roseman, J. J. [1967]: The principle of Saint-Venant in linear and nonlinear plane elasticity, Arch. Rational Mech. Anal. 25, 142–162.
Stein, E. M. [1970]: Singular integrals and differentiability properties of functions Princeton Univ. Press.
De Saint-Venant, B. [1856]: Mémoire sur la torsion des prismes, Mémoires présentés pars divers savants à l'académie de sciences de l'institut impérial de France, 2. Ser. 14, 233–560.
Toupin, R. A. [1965]: Saint-Venant's principle, Arch. Rational Mech. Anal. 18, 83–96.
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Mielke, A. Normal hyperbolicity of center manifolds and Saint-Venant's principle. Arch. Rational Mech. Anal. 110, 353–372 (1990). https://doi.org/10.1007/BF00393272
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DOI: https://doi.org/10.1007/BF00393272