1Universidad Nacional de Colombia, Bogotá, Colombia. Email: feponcev@unal.edu.co
2Universidad Nacional de Colombia, Bogotá, Colombia. Email: llebedev@unal.edu.co
3Universidad Nacional de Colombia, Bogotá, Colombia. Email: lrendona@unal.edu.co ]]>
This paper concerns with existence and uniqueness of a weak solution for elliptic systems of partial differential equations with mixed boundary conditions. The proof is based on establishing the coerciveness of bilinear forms, related with the system of equations, which depend on first-order derivatives of vector functions in Rn. The condition of coerciveness relates to Korn's type inequalities. The result is illustrated by an example of boundary value problems for a class of elliptic equations including the equations of linear elasticity.
Key words: Weak solvability, Boundary value problems, Elliptic equations, Korn's type inequality.
Este artículo trata sobre la existencia y unicidad de una solución débil para sistemas elípticos de ecuaciones diferenciales parciales con condiciones de frontera mixtas. La demostración se basa en la determinación de la coercividad de formas bilineales, relacionadas con el sistema de ecuaciones, las cuales dependen de las derivadas de primer orden de funciones vectoriales en Rn. La condición de coercividad se relaciona con desigualdades tipo Korn. El resultado se ilustra mediante un ejemplo de problemas con valores en la frontera para una clase de ecuaciones elípticas, incluyendo las ecuaciones de elasticidad lineal.
Palabras clave: Solubilidad débil, problemas con valores en la frontera, ecuaciones elípticas, desigualdad tipo Korn.
Texto completo disponible en PDF
[1] G. Acosta, R. G. Durán, and F. L. García, `Korn Inequality and Divergence Operator: Counterexamples and Optimality of Weighted Estimates´, Proc. Amer. Math. Soc. 141, (2013), 217-232. [ Links ]
[2] R. A. Adams and J. F. Fournier, Sobolev Spaces, 2nd edn, Elsevier, Amsterdam, The Netherlands, 2003. [ Links ]
[3] H. Altenbach, V. A. Eremeyev, and L. P. Lebedev, `On the Existence of Solution in the Linear Elasticity with Surface Stresses´, ZAMM 90, (2010), 231-240. [ Links ]
[4] H. Altenbach, V. A. Eremeyev, and L. P. Lebedev, `On the Spectrum and Stiffness of an Elastic Body with Surface Stresses´, ZAMM 91, (2011), 699-710. [ Links ]
[5] P. G. Ciarlet, Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity, Vol. 1, Elsevier, Amsterdam, The Netherlands, 1988. [ Links ]
[6] S. Conti, D. Faraco, and F. Maggi, `A New Approach to Counterexamples to L1 Estimates: Korn's Inequality, Geometric Rigidity, and Regularity for Gradients of Separately Convex Functions´, Arch. Rational Mech. Anal. 175, (2005), 287-300. [ Links ]
[7] H. L. Duan, J. Wang, and B. L. Karihaloo, `Theory of Elasticity at the Nanoscale´, Advances in Applied Mechanics 42, (2009), 1-68. [ Links ]
[8] G. Fichera, Handbuch der Physik, Springer, Berlin, Germany, (1972), chapter Existence Theorems in Elasticity. [ Links ]
[9] K. O. Friederichs, `On the Boundary-Value Problems of the Theory of Elasticity and Korn's Inequality´, Ann. of Math. 48, (1947), 441-471. [ Links ]
[10] M. E. Gurtin and A. I. Murdoch, `A Continuum Theory of Elastic Material Surfaces´, Arch. Rat. Mech. Analysis 57, (1975), 291-323. [ Links ]
[11] C. O. Horgan, `Korn's Inequalities and Their Applications in Continuum Mechanics´, SIAM rev. 37, 4 (1995), 491-511. [ Links ]
[12] V. A. Kondrat'ev and O. A. Oleynik, `Boundary-value Problems for the System of Elasticity Theory in Unbounded Domains. Korn's Inequalities´, Russ. Math. Surv. 43, 5 (1988), 65-119. [ Links ]
[13] L. P. Lebedev, I. I. Vorovich, and M. J. Cloud, Functional Analysis in Mechanics, 2nd edn, Springer-Verlag, New York, USA, 2013. [ Links ]
[14] S. G. Mikhlin, The Problem of the Minimum of a Quadratic Functional, Governmental publisher, Moscow, Rusia, 1952. in russian [ Links ]
[15] J. Necas and I. Hlavácek, Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Elsevier, 1981. [ Links ]
]]>[16] I. I. Vorovich, Nonlinear Theory of Shallow Shells, Springer-Verlag, New York, USA, 1999. [ Links ]
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCMv47n2a06,
AUTHOR = {Ponce, Felipe and Lebedev, Leonid and Rendón, Leonardo},
TITLE = {{On Weak Solvability of Boundary Value Problems for Elliptic Systems}},
JOURNAL = {Revista Colombiana de Matemáticas},
YEAR = {2013},
volume = {47},
number = {2}, ]]>
pages = {191--204}
}