A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems

Abstract

For scalar variational problems subject to linear boundary values, we determine completely those integrands W : ℝ n → ℝ for which the minimum is not attained, thereby completing previous efforts such as a recent nonexistence theorem of Chipot [ 9 ] and unifying a large number of examples and counterexamples in the literature. As a corollary, we show that in case of nonattainment (and provided W grows superlinearly at infinity), every minimising sequence converges weakly but not strongly in W 1,1 (Ω) to a unique limit, namely the linear deformation prescribed at the boundary, and develops fine structure everywhere in Ω, that is to say every Young measure associated with the sequence of its gradients is almost-nowhere a Dirac mass. Connections with solid–solid phase transformations are indicated.