Ferrel, Juan Limaco; Medeiros, Luis Adauto Vibrations of elastic membranes with moving boundaries. (English) Zbl 1008.74039 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 45, No. 3, 363-382 (2001). The authors investigate a mathematical model for small deformations of a circular membrane when the boundary is not fixed and can move. The theory is based on a wave equation of the form \({\partial^2 u\over \partial t^2}-[a(t)+ b(t)\int_{\Omega_t} |\nabla u(x,t)|^2 dx] \Delta u=0\). Reviewer: J.Genin (Las Cruces) Cited in 5 Documents MSC: 74H45 Vibrations in dynamical problems in solid mechanics 74K15 Membranes Keywords:moving boundary; small deformations; circular membrane; wave equation PDFBibTeX XMLCite \textit{J. L. Ferrel} and \textit{L. A. Medeiros}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 45, No. 3, 363--382 (2001; Zbl 1008.74039) Full Text: DOI References: [1] H. Bresis, Analyse Fonctionelle (Théorie et Applications), Masson, Paris, 1983.; H. Bresis, Analyse Fonctionelle (Théorie et Applications), Masson, Paris, 1983. [2] G.F. Carrier, On the nonlinear vibration problem of the elastic string, Q. J. Appl. Math., 3 (1945) 157-165.; G.F. Carrier, On the nonlinear vibration problem of the elastic string, Q. J. Appl. Math., 3 (1945) 157-165. · Zbl 0063.00715 [3] G. Kirchhoff, Vorlesungen über Mechanik, Tauber, Leipzig, 1883.; G. Kirchhoff, Vorlesungen über Mechanik, Tauber, Leipzig, 1883. [4] J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, Dunod, Paris, 1969.; J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, Dunod, Paris, 1969. [5] Limaco, J.; Medeiros, L. A., Kirchhoff-Carrier elastic strings in noncylindrical domains, in Portuga. Mat., 56, 4, 465-500 (1999) · Zbl 0943.35001 [6] L.A. Medeiros, J. Limaco Ferrel, S.B. Menezes, Vibrations of Elastic Strings (Mathematical Aspects), Atas do \(48^°\); L.A. Medeiros, J. Limaco Ferrel, S.B. Menezes, Vibrations of Elastic Strings (Mathematical Aspects), Atas do \(48^°\) [7] Simon, J., Compact Sets in Space \(L^p(0,T;B)\), Ann. Mat. Pure Appl., 146, 4, 65-96 (1987) · Zbl 0629.46031 [8] R. Temam, Navier-Stokes Equations (Theory and Numerical Analysis), North-Holland, Amsterdam, 1979.; R. Temam, Navier-Stokes Equations (Theory and Numerical Analysis), North-Holland, Amsterdam, 1979. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.