Abstract
We establish existence and uniqueness of solutions of a class of partial differential equations with nonlocal Dirchlet conditions in weighted function spaces. The problem is motivated by the study of the probability distribution of the response of an elasto-plastic oscillator when subjected to white noise excitation (see [1,2] on the derivation of the boundary value problem). Note that the developments in [1,2] are based on an extension of Khasminskii’s method (see, e.g. [5]) and in this paper we use a direct approach to achieve our objectives.
We refer the reader to [3, 4, 6, 7] for general background on modeling, theoretical, and computational issues related to elasto-plastic oscillators.
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References
A. Bensoussan and J. Turi. Stochastic variational inequalities for elasto-plastic oscillators. C. R. Math. Acad. Sci. Paris, 343(6):399–406, 2006.
A. Bensoussan and J. Turi. Degenerate Dirichlet problems related to the invariant measure of elasto-plastic oscillators. Appl. Math. Optim., 58(1):1–27, 2007. DOI 10.1007/s00245-007-9027-4 (available online).
C. Féau. Les méthodes probabilistes en mécanique sismique. Application aux calculs de tuyauteries fissurées. Thèse, Université d’Evry, 2003.
D. Karnopp and T. D. Scharton. Plastic deformation in random vibration. J. Acoust. Soc. Amer., 39:1154–1161, 1966.
R. Z. Khasminskii. Stochastic stability of differential equations. Sijthoff and Noordhoff, Alphen aan den Rijn, 1980.
A. Preumont. Random vibration and spectral analysis. Kluwer Academic Publ., Dordrecht, 2nd edition, 1994.
J. B. Roberts and P.-T. D. Spanos. Random vibration and statistical linearization. John Wiley & Sons, Chichester, 1990.
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Bensoussan, A., Turi, J. (2010). On a Class of Partial Differential Equations with Nonlocal Dirichlet Boundary Conditions. In: Fitzgibbon, W., Kuznetsov, Y., Neittaanmäki, P., Périaux, J., Pironneau, O. (eds) Applied and Numerical Partial Differential Equations. Computational Methods in Applied Sciences, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3239-3_3
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DOI: https://doi.org/10.1007/978-90-481-3239-3_3
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