Rank-one convex functions on 2×2 symmetric matrices and laminates on rank-three lines

We construct a function on the space of symmetric 2× 2 matrices in such a way that it is convex on rank-one directions and its distributional Hessian is not a locally bounded measure. This paper is also an illustration of a recently proposed technique to disprove L¹ estimates by the construction of suitable probability measures (laminates) in matrix space. From this point of view the novelty is that the support of the laminate, besides satisfying a convex constraint, needs to be contained on a rank-three line, up to arbitrarily small errors. Mathematics Subject Classification (2000): 35J50 – 26D15 – 49N60 –74G65