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.

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