Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

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

Gero Frieseckea1 p1

a1 Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, U.K.

Abstract

For scalar variational problems

S0308210500028730_eqnU1

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 [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 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)

Correspondence

p1 Current address: Department of Mathematics and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, PA 152113, U.S.A.