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Rank-one convex functions on 2x2 symmetric matrices and laminates on rank-three lines
Sergio Conti, Daniel Faraco, Francesco Maggi and Stefan Müller
We construct a function on the space of symmetric $2\times 2$ 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 $L^1$ 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.