On the positivity of the second variation in finite elasticity

Abstract

where f is a vector-valued function on a bounded open domain O (1%" that satisfies f = d (d given) on a portion of the boundary 9 . Here (Vf) e = OfffSxj, i , j = 1, 2, . . . , n. I f f is a sufficiently smooth solution of the corresponding Euler-Lagrange equations, then a further necessary condition for it to be a weak relative minimizer (a local minimizer in the Cx(~) topology) is that the second variation of I at f be nonnegative2'3; i.e., ~IAu) := f Vu. G[Vu] _> o I2