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)

 
 

 

On a stochastic wave equation with unilateral boundary conditions


Author: Jong Uhn Kim
Journal: Trans. Amer. Math. Soc. 360 (2008), 575-607
MSC (2000): Primary 35L65, 35R60, 60H15
DOI: https://doi.org/10.1090/S0002-9947-07-04143-8
Published electronically: July 20, 2007
MathSciNet review: 2346463
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the existence and uniqueness of solutions to the initial boundary value problem for a one-dimensional wave equation with unilateral boundary conditions and random noise. We also establish the existence of an invariant measure.


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

  • [1] Da Prato, G. and Zabczyk, J., Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 1992. MR 1207136 (95g:60073)
  • [2] Da Prato, G. and Zabczyk, J., Ergodicity for infinite dimensional systems, Cambridge University Press, Cambridge, 1996. MR 1417491 (97k:60165)
  • [3] Do, C., On the dynamic deformation of a bar against an obstacle, in ``Variational methods in the Mechanics of Solids" (1980), pp. 237 -241.
  • [4] Duvaut, G. and Lions, J.L., Inequalities in Mechanics and Physics, Springer, New York-Berlin, 1976. MR 0521262 (58:25191)
  • [5] Hausmann, U.G. and Pardoux, E., Stochastic variational inequalities of parabolic type, Applied Math. Opt. 20 (1989), pp. 163 - 193. MR 0998402 (90k:60119)
  • [6] John, F., Partial Differential Equations, 4th edition, Springer-Verlag, New York-Heidelberg-Berlin, 1982. MR 0831655 (87g:35002)
  • [7] Kalker, J., Aspects of contact mechanics, in ``The Mechanics of the Contact between Deformable Bodies" (1975), pp. 1 - 25.
  • [8] Karatzas, I. and Shreve, S., Brownian motion and Stochastic Calculus, 2nd edition, Springer, New York-Berlin-Heidelberg, 1997. MR 0917065 (89c:60096)
  • [9] Kim, J.U., A one-dimensional dynamic contact problem in linear viscoelasticity, Math. Methods Appl. Sci. 13 (1990), pp. 55 - 79. MR 1060224 (91f:35182)
  • [10] Kim, J.U., Invariant measures for the stochastic von Karman plate equation, SIAM J. Math. Anal. 36 (2005), pp. 1689 - 1703. MR 2139567 (2006c:60073)
  • [11] Lions, J.L., Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969.MR 0259693 (41:4326)
  • [12] Schatzman, M., A hyperbolic problem of second order with unilateral constraints, J. Math. Anal. Appl. 73 (1980), pp. 138 - 191. MR 0560941 (81d:35047)
  • [13] Schatzman, M., The penalty method for the vibrating string with an obstacle, in ``Analytical and Numerical Approaches to Asymptotic Problems in Analysis" (1981), pp. 345 - 357. MR 0605520 (82i:35124)
  • [14] Tulcea, A.I. and Tulcea, C.I., Topics in the Theory of Lifting, Springer-Verlag, New York, 1969. MR 0276438 (43:2185)
  • [15] Villaggio, P., A unilateral contact problem in linear elasticity, J. Elasticity 10 (1980), pp. 113 - 119. MR 0576162 (81d:73099)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35L65, 35R60, 60H15

Retrieve articles in all journals with MSC (2000): 35L65, 35R60, 60H15


Additional Information

Jong Uhn Kim
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123
Email: kim@math.vt.edu

DOI: https://doi.org/10.1090/S0002-9947-07-04143-8
Keywords: Unilateral boundary conditions, Brownian motion, existence of a solution, pathwise uniqueness, invariant measure, probability distribution.
Received by editor(s): June 9, 2004
Received by editor(s) in revised form: July 17, 2005
Published electronically: July 20, 2007
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society