a1 Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, U.K.
For scalar variational problems
subject to linear boundary values, we determine completely those integrands W: ℝn → ℝ for which the minimum is not attained, thereby completing previous efforts such as a recent nonexistence theorem of Chipot  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 W1,1(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.
(Received October 21 1992)
(Revised March 14 1993)
p1 Current address: Department of Mathematics and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, PA 152113, U.S.A.