Journal of the Mathematical Society of Japan

Linear approximation for equations of motion of vibrating membrane with one parameter

Koji KIKUCHI

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Abstract

This article treats a one parameter family of equations of motion of vibrating membrane whose energy functionals converge to the Dirichlet integral as the parameter ε tends to zero. It is proved that both weak solutions satisfying energy inequality and generalized minimizing movements converge to a unique solution to the d’Alembert equation.

Article information

Source
J. Math. Soc. Japan, Volume 60, Number 1 (2008), 127-169.

Dates
First available in Project Euclid: 24 March 2008

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1206367958

Digital Object Identifier
doi:10.2969/jmsj/06010127

Mathematical Reviews number (MathSciNet)
MR2392006

Zentralblatt MATH identifier
1142.35049

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations
Secondary: 49J40: Variational methods including variational inequalities [See also 47J20] 49Q15: Geometric measure and integration theory, integral and normal currents [See also 28A75, 32C30, 58A25, 58C35]

Keywords
hyperbolic equations linear approximation BV functions minimizing movements varifolds

Citation

KIKUCHI, Koji. Linear approximation for equations of motion of vibrating membrane with one parameter. J. Math. Soc. Japan 60 (2008), no. 1, 127--169. doi:10.2969/jmsj/06010127. https://projecteuclid.org/euclid.jmsj/1206367958


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References

  • [1] \auF. Almgren, J. E. Taylor and L. Wang, Curvature driven flows: a variational approach, \tiSIAM J. Control. and Optim., , 31 ((1993),)\spg387–\epg438.
  • [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Science Publication, 2000.
  • [3] F. Bethuel, J. M. Coron, J. M. Ghidallia and A. Soyeur, Heat flow and relaxed energies for harmonic maps, Nonlinear Diffusion Equations and Their Equilibrium States, 3.
  • [4] L. C. Evans, Weak convergence methods for nonlinear partial differential equations, CBMS, 74, Amer. Math. Soc., 1990.
  • [5] L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, CRC Press, 1992.
  • [6] H. Federer, Geometric measure theory, Springer-Verlag, New York, 1969.
  • [7] \auD. Fujiwara and S. Takakuwa, A varifold solution to the nonlinear equation of motion of a vibrating membrane, Kodai Math. J., 9 (1986), 84–116, correction, \tiibid., , 14 ((1991),)\spg310–\epg311.
  • [8] M. Giaquinta, G. Modica and J. Soucek, Cartesian currents in the calculus of variations I, II, Springer, 1998.
  • [9] E. De Giorgi, New problems on minimizing movements, Boundary Value Problems for PDE and Applications, Masson, 1993, pp. 81–98.
  • [10] E. Giusti, Minimal surfaces and functions of bounded variation, Birkhäuser, Boston-Basel-Stuttgart, 1984.
  • [11] \auK. Kikuchi, An analysis of the nonlinear equation of motion of a vibrating membrane in the space of $\mathit{BV}$ functions, \tiJ. Math. Soc. Japan, , 52 ((2000),)\spg741–\epg766.
  • [12] \auK. Kikuchi, A remark on Dirichlet boundary condition for the nonlinear equation of motion of a vibrating membrane, \tiNonlinear Analysis, , 47 ((2001),)\spg1039–\epg1050.
  • [13] N. Kikuchi, An approach to the construction of Morse flows for variational functionals, Nematics Mathematical and Physical Aspects (eds. J. M. Ghidaglia, J. M. Coron and F. Hélein), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 332, Kluwer Acad. Publ., Dordrecht-Boston-London, 1991, pp. 195–199.
  • [14] \auD. Kinderlehrer and P. Pedregal, Gradient young measures generated by sequences in sobolev spaces, \tiJ. Geom. Anal., , 4 ((1994),)\spg59–\epg90.
  • [15] T. Nagasawa, Discrete Morse semiflows and evolution equations, Proceedings of the 16th Young Japanese Mathematicians' Seminar on Evolution Equations, 1994, pp. 1–20.
  • [16] T. Nagasawa, Construction of weak solutions of the navier-stokes equations on riemannian manifold by minimizing variational functionals, Adv. Math. Sci. Appl., (1998).
  • [17] \auK. Rektorys, On application of direct variational method to the solution of parabolic boundary value problems of arbitrary order in the space variables, \tiCzechoslovak Math. J., , 21 ((1971),)\spg318–\epg339.
  • [18] \auE. Rothe, Zweidimensionale parabolische randwertaufgaben als grenzfall eindimensionaler randwertaufgaben, \tiMath. Ann., , 102 ((1930),)\spg650–\epg670.
  • [19] \auA. Tachikawa, A variational approach to constructing weak solutions of semilinear hyperbolic systems, \tiAdv. Math. Sci. Appl., , 3 ((1994),)\spg93–\epg103.