Direct Method of Calculus of Variations in Elasticity

International audienceDEFINITIONTheorems on the existence of minimizers for functionals defined on Banach spaces, and related approximation methods, applied to minimization problems arising in elasticity theory.INTRODUCTIONVariational methods are a powerful tool in elasticity and in fact the only known approach able to guarantee sufficient generality in the treatment of problems arising in hyperelasticity (see the corresponding entry), in the asymptotic derivation of two-dimensional elastic models (Ciarlet, 1988, 1997), and in many other cases in continuum mechanics (Pedregal, 2000; Fonseca, 1987; dell’Isola and Placidi, 2011). Variational problems take usually the form of a minimization problem that can be described as follows: we search the minimum of a functional F(u) defined on a subset S of a Banach space B (i.e., a normed, complete vector space) and taking values in [−∞, +∞]. Direct method provides sufficient conditions on S, B, and F for the existence of a minimizer ũ∈