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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
50/2004

Rank-one convex functions on 2x2 symmetric matrices and laminates on rank-three lines

Sergio Conti, Daniel Faraco, Francesco Maggi and Stefan Müller

Abstract

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.

Received:
Aug 13, 2004
Published:
Aug 13, 2004

Related publications

inJournal
2005 Repository Open Access
Sergio Conti, Daniel Faraco, Francesco Maggi and Stefan Müller

Rank-one convex functions on 2x2 symmetric matrices and laminates on rank-three lines

In: Calculus of variations and partial differential equations, 24 (2005) 4, pp. 479-493