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

Under mild conditions on a polyconvex function

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

\phi: {\bf R}^{5} \to {{\bf R}} ϕ : R 5 → R

satisfying

\phi(\xi,\det \xi) \leq W(\xi) ϕ ( ξ , det ⁡ ξ ) ≤ W ( ξ )

for all

2 \times 2 2 × 2

matrices

\xi ξ

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

W: {\bf R}^{2 \times 2} \to {{\bf R}} W : R 2 × 2 → R

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

\phi ϕ

as above and which also satisfy

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

for at least one

2 \times 2 2 × 2

matrix

\xi_{0} ξ 0