A Remark on the Structure of the Busemann Representative of a Polyconvex Function

Abstract

Under mild conditions on a polyconvex function

W: \R^{2 \times 2} \to \R

, its largest convex representative, known as the Busemann representative, may be written as the supremum over all affine functions

\phi: \R^{5} \to \R

satisfying

\phi(\xi,\det \xi) \leq W(\xi)

for all

2 \times 2

matrices

\xi

. In this paper, we construct an example of a polyconvex

W: \R^{2 \times 2} \to \R

whose Busemann representative is, on an open set, strictly larger than the supremum of all affine functions

\phi

as above and which also satisfy

\phi(\xi_{0},\det \xi_{0}) = W(\xi_{0})

for at least one

2 \times 2

matrix

\xi_{0}

Contact details are reproduced from the original publication and may be historical.

Dept. of Mathematics, University of Surrey, Guildford, GU2 7XH, United Kingdom

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press.