Decay estimates for the biharmonic equation with applications to Saint-Venant principles in plane elasticity and Stokes flows

Author:
C. O. Horgan

Journal:
Quart. Appl. Math. **47** (1989), 147-157

MSC:
Primary 73C02; Secondary 31A30, 35B40, 73C10, 76D05

DOI:
https://doi.org/10.1090/qam/987903

MathSciNet review:
987903

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**[1]**James K. Knowles,*An energy estimate for the biharmonic equation and its application to Saint-Venant’s principle in plane elastostatics*, Indian J. Pure Appl. Math.**14**(1983), no. 7, 791–805. MR**714832****[2]**Cornelius O. Horgan and James K. Knowles,*Recent developments concerning Saint-Venant’s principle*, Adv. in Appl. Mech.**23**(1983), 179–269. MR**889288****[3]**M. E. Gurtin,*The linear theory of elasticity, Handbuch der Physik*, S. Flügge and C. Truesdell (eds.), Vol. 6a/2, Springer-Verlag, Berlin, 1972, pp. 1-295**[4]**C. O. Horgan,*Plane entry flows and energy estimates for the Navier-Stokes equations*, Arch. Rational Mech. Anal.**68**(1978), no. 4, 359–381. MR**0521600**, https://doi.org/10.1007/BF00250987**[5]**James K. Knowles,*On Saint-Venant’s principle in the two-dimensional linear theory of elasticity*, Arch. Rational Mech. Anal.**21**(1966), 1–22. MR**0187480**, https://doi.org/10.1007/BF00253046**[6]**R. A. Toupin,*Saint-Venant’s principle*, Arch. Rational Mech. Anal.**18**(1965), 83–96. MR**0172506**, https://doi.org/10.1007/BF00282253**[7]**J. N. Flavin,*On Knowles’ version of Saint-Venant’s principle in two-dimensional elastostatics*, Arch. Rational Mech. Anal.**53**(1973/74), 366–375. MR**0337090**, https://doi.org/10.1007/BF00281492**[8]**O. A. Oleĭnik and G. A. Iosif′jan,*The Saint-Venant principle in plane elasticity theory*, Dokl. Akad. Nauk SSSR**239**(1978), no. 3, 530–533 (Russian). MR**0502616****[9]**O. A. Oleinik and G. A. Yosifian,*The Saint-Venant principle in the two-dimensional theory of elasticity and boundary problems for a biharmonic equation in unbounded domains*, Sibirsk. Mat. Zh.**19**, 1154-1165 (1978) (translated in Siberian Math. J.**19**, 813-822 (1978))**[10]**O. A. Oleĭnik,*Energetic estimates analogous to the Saint-Venant principle and their applications*, Equadiff IV (Proc. Czechoslovak Conf. Differential Equations and their Applications, Prague, 1977) Lecture Notes in Math., vol. 703, Springer, Berlin, 1979, pp. 328–339. MR**535353****[11]**O. A. Oleinik,*Applications of the energy estimates analogous to Saint-Venant's principle to problems of elasticity and hydrodynamics*, Lecture Notes in Phys.**90**, 422-432 (1979)**[12]**J. L. Ericksen,*Special topics in elastostatics, Advances in Applied Mechanics*, C. S. Yih (ed.), Vol. 17, Academic Press, New York, 1977, pp. 189-244**[13]**R. G. Muncaster,*Saint-Venant’s problem in nonlinear elasticity: a study of cross sections*, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 17–75. MR**584396****[14]**G. H. Hardy, J. E. Littlewood, and G. Pólya,*Inequalities*, 2nd ed., Cambridge University Press, Cambridge, 1967**[15]**S. G. Mikhlin,*Variational methods in mathematical physics*, Translated by T. Boddington; editorial introduction by L. I. G. Chambers. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR**0172493****[16]**Cornelius O. Horgan,*A note on a class of integral inequalities*, Proc. Cambridge Philos. Soc.**74**(1973), 127–131. MR**0331152****[17]**J. N. Flavin and R. J. Knops,*Some convexity considerations for a two-dimensional traction problem*, Z. Angew. Math. Phys.**39**(1988), no. 2, 166–176. MR**937701**, https://doi.org/10.1007/BF00945763**[18]**P. Vafeades and C. O. Horgan,*Exponential decay estimates for solutions of the von Kármán equations on a semi-infinite strip*, Arch. Rational Mech. Anal.**104**(1988), no. 1, 1–25. MR**956565**, https://doi.org/10.1007/BF00256930

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DOI:
https://doi.org/10.1090/qam/987903

Article copyright:
© Copyright 1989
American Mathematical Society