Calculus of Variations and Geometric Measure Theory

S. Conti - D. Faraco - F. Maggi - S. Müller

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

created on 17 Aug 2004
modified by maggi on 14 Dec 2006


Published Paper

Inserted: 17 aug 2004
Last Updated: 14 dec 2006

Journal: Calc. Var. Partial Differential Equations
Volume: 24
Number: 4
Pages: 479-493
Year: 2005

Preprint MPI-MIS 502004


We construct a function on the space of 2x2 symmetric 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.

Keywords: bounded variation, convex integration, rank-one convexity, laminates