The energy method for elastic problems with non-homogeneous boundary conditions

Ramon Quintanilla

International Journal of Applied Mathematics and Computer Science (2002)

• Volume: 12, Issue: 1, page 91-100
• ISSN: 1641-876X

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In this paper we propose the weighted energy method as a way to study estimates of solutions of boundary-value problems with non-homogeneous boundary conditions in elasticity. First, we use this method to study spatial decay estimates in two-dimensional elasticity when we consider non-homogeneous boundary conditions on the boundary. Some comments in the case of harmonic vibrations are considered as well. We also extend the arguments to a class of three-dimensional problems in a cylinder. A section is devoted to the study of an ill-posed problem. Some remarks are presented in the last section of the paper.

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Quintanilla, Ramon. "The energy method for elastic problems with non-homogeneous boundary conditions." International Journal of Applied Mathematics and Computer Science 12.1 (2002): 91-100. <http://eudml.org/doc/207572>.

@article{Quintanilla2002,
abstract = {In this paper we propose the weighted energy method as a way to study estimates of solutions of boundary-value problems with non-homogeneous boundary conditions in elasticity. First, we use this method to study spatial decay estimates in two-dimensional elasticity when we consider non-homogeneous boundary conditions on the boundary. Some comments in the case of harmonic vibrations are considered as well. We also extend the arguments to a class of three-dimensional problems in a cylinder. A section is devoted to the study of an ill-posed problem. Some remarks are presented in the last section of the paper.},
author = {Quintanilla, Ramon},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {weighted energy method; Navier equations; decay estimates; non-homogeneous boundary conditions; strip; three-dimensional elasticity; boundary value problems; nonhomogeneous boundary conditions; spatial decay estimates; two-dimensional elasticity; harmonic vibrations; cylinder; ill-posed problem},
language = {eng},
number = {1},
pages = {91-100},
title = {The energy method for elastic problems with non-homogeneous boundary conditions},
url = {http://eudml.org/doc/207572},
volume = {12},
year = {2002},
}

TY - JOUR
AU - Quintanilla, Ramon
TI - The energy method for elastic problems with non-homogeneous boundary conditions
JO - International Journal of Applied Mathematics and Computer Science
PY - 2002
VL - 12
IS - 1
SP - 91
EP - 100
AB - In this paper we propose the weighted energy method as a way to study estimates of solutions of boundary-value problems with non-homogeneous boundary conditions in elasticity. First, we use this method to study spatial decay estimates in two-dimensional elasticity when we consider non-homogeneous boundary conditions on the boundary. Some comments in the case of harmonic vibrations are considered as well. We also extend the arguments to a class of three-dimensional problems in a cylinder. A section is devoted to the study of an ill-posed problem. Some remarks are presented in the last section of the paper.
LA - eng
KW - weighted energy method; Navier equations; decay estimates; non-homogeneous boundary conditions; strip; three-dimensional elasticity; boundary value problems; nonhomogeneous boundary conditions; spatial decay estimates; two-dimensional elasticity; harmonic vibrations; cylinder; ill-posed problem
UR - http://eudml.org/doc/207572
ER -

References

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2. Flavin J.N., Knops R.J. and Payne L.E. (1989): Decay estimates for the constrained elastic cylinder of variable cross section. - Quart. Appl. Math., Vol. XLVII, No. 2, pp. 325-350. Zbl0706.73015
3. Franchi F. and Straughan B. (1994): Spatial decay estimates and continuous dependence on modelling for an equation from dynamotheory. - Proc. Roy. Society. Lond. A, Vol. 445, pp. 437-451. Zbl0809.35150
4. Galdi G. and Rionero S. (1985): Weigthed Energy Methods in Fluid Dynamics and Elasticity. -Berlin: Springer. Zbl0585.76001
5. Horgan C.O.(1989): Recent developments concerning Saint-Venant's principle: An update. - Appl. Mech. Rev., Vol. 42, No. 11, pp. 295-303. 
6. Horgan C.O.(1996): Recent developments concerning Saint-Venant's principle: A second update. - Appl. Mech. Rev., Vol. 49, No. 10, pp. 101-111. 
7. Horgan C.O. and Knowles J.K. (1983): Recent developments concerning Saint-Venant's principle, In: Advances in Applied Mechanics (J.W. Hutchinson, Ed.). - New York: Academic Press, pp. 179-269. Zbl0569.73010
8. Horgan C.O. and Payne L.E. (1992): The influence of geometric perturbations on the decay of Saint-Venantend effects in linear isotropic elasticity, In: Partial Differential Equations with Real Analysis (H. Begrehr and A. Jeffrey, Eds.). - Essex: Longman, pp. 187-218. Zbl0786.73015
9. Knops R.J. and Payne L.E. (1998): Spatial behaviour of energy in partially constrained thick elasticplates. - Atti dei Convegni Lincei, Vol. 140, pp. 77-104. 
10. Lin C. and Payne L.E. (1993): On the spatial decay of ill-posed parabolic problems. - Math. Mod. Meth. Appl. Sci., Vol. 3, No. 4, pp. 563-575. Zbl0817.35034
11. Quintanilla R. (1997a): Spatial decay estimates and upper bounds in elasticity for domains with unbounded cross-sections. - J. Elasticity, Vol. 46, No. 3, pp. 239-254. Zbl0884.73008
12. Quintanilla R. (1997b): Directions of spatial decay in linear elasticity. - Manuscript (unpublished). 
13. Quintanilla R. (1998): Comportamiento espacial en sólidos elasticos noacotados. - Anales de Ingenieria Mecanica, Vol.12, No. 1, pp.175-180. 
14. Quintanilla R. (2000): Energy methods for problems with non homogeneous boundaryconditions. - Manuscript (unpublished). 
15. Straughan B. (1982): Instability, Nonexistence and Weighted Energy Methods in Fluid Dynamics and Related Theories. -London: Longman. Zbl0492.76001
16. Weinberger H.F. (1995): A First Course in Partial Differential Equations. -New York: Dover. 

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