Oscillations in one-dimensional elasticity with surface energy
Authors:
Irene Fonseca, Jack Schaeffer and Mikhail M. Shvartsman
Journal:
Quart. Appl. Math. 57 (1999), 475-499
MSC:
Primary 74N15; Secondary 35B35, 35Q72, 74B20, 74D10, 74G65
DOI:
https://doi.org/10.1090/qam/1704443
MathSciNet review:
MR1704443
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The characterization of the oscillatory behavior of solutions of a semilinear equation in one space dimension is obtained. In this work the model equation for a material undergoing a phase transition encompasses a surface energy term and first-order memory effects.
J. P. Aubin, Un théorème de compacité, C. R. Acad. Sci. 256, 5042–5044 (1963)
- J. M. Ball, P. J. Holmes, R. D. James, R. L. Pego, and P. J. Swart, On the dynamics of fine structure, J. Nonlinear Sci. 1 (1991), no. 1, 17–70. MR 1102830, DOI https://doi.org/10.1007/BF01209147
- I. Fonseca, D. Brandon, and P. Swart, Dynamics and oscillatory microstructure in a model of displacive phase transformations, Progress in partial differential equations: the Metz surveys, 3, Pitman Res. Notes Math. Ser., vol. 314, Longman Sci. Tech., Harlow, 1994, pp. 130–144. MR 1316196
L. C. Evans, Partial Differential Equations, Berkeley Mathematics Lecture Notes, 1994
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
- Lamberto Cesari (ed.), Contributions to modern calculus of variations, Pitman Research Notes in Mathematics Series, vol. 148, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. Papers from the symposium marking the centenary of the birth of Leonida Tonelli held in Bologna, May 13–14, 1985. MR 894067
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
- Robert L. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability, Arch. Rational Mech. Anal. 97 (1987), no. 4, 353–394. MR 865845, DOI https://doi.org/10.1007/BF00280411
- L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
L. Tartar, Nonlinear partial differential equations using compactness method, Report 1584, MRC, Univ. Wisconsin, 1975.
L. Truskinovsky and G. Zanzotto, Ericksen’s bar revisited, Preprint, 1994
J. P. Aubin, Un théorème de compacité, C. R. Acad. Sci. 256, 5042–5044 (1963)
J. M. Ball, P. J. Holmes, R. D. James, R. L. Pego, and P. J. Swart, On the dynamics of fine structure, J. Nonlinear Science 1, 17–70 (1991)
D. Brandon, I. Fonseca, and P. J. Swart, The creation and propagation of oscillatory microstructure in a dynamical model of displacive phase transformations, in Progress in Partial Differential Equations: the Metz Surveys 1994, 3, Pitman Res. Notes Math. Ser. 314, Longman Sci. Tech., Harlow, 1994, pp. 130–144
L. C. Evans, Partial Differential Equations, Berkeley Mathematics Lecture Notes, 1994
J. L. Lions, Quelques Méthodes de Résolution de Problèmes aux Limites non Linéaires, Dunod Gauthier-Villars, 1969
F. Murat, A survey on compensated compactness, in Contributions to Modern Calculus of Variations, (ed. L. Cesari), Pitman Research Notes in Math., Longman, Harlow, 1987, pp. 145–183
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983
R. L. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility and stability, Arch. Rational Mech. Anal. 97, 353–394 (1987)
L. Tartar, The compensated compactness and applications to PDEs, in Nonlinear Analysis and Mechanics (ed. R. Knops), Pitman Research Notes in Math., Vol. 4, Pitman, London-San Francisco, Longman, Harlow, 1979, pp. 136–212
L. Tartar, Nonlinear partial differential equations using compactness method, Report 1584, MRC, Univ. Wisconsin, 1975.
L. Truskinovsky and G. Zanzotto, Ericksen’s bar revisited, Preprint, 1994
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
74N15,
35B35,
35Q72,
74B20,
74D10,
74G65
Retrieve articles in all journals
with MSC:
74N15,
35B35,
35Q72,
74B20,
74D10,
74G65
Additional Information
Article copyright:
© Copyright 1999
American Mathematical Society
