On Rank One Convex Functions that are Homogeneous of Degree One

By B. Kirchheim, Jan Kristensen. 2015

120 citations recorded by the source corpus.

We show that positively 1-homogeneous rank one convex functions are convex at 0 and at matrices of rank one. The result is a special case of an abstract convexity result that we establish for positively 1-homogeneous directionally convex functions defined on an open convex cone in a finite dimensional vector space. From these results we derive a number of consequences including various generalizations of the Ornstein L1 non inequalities. Most of the results were announced in (C R Acad Sci Paris Ser I 349:407–409,2011).

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/e22b768db2ea7ca3d2b82d4229561e9378b7c01d

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

On the structure of A-free measures and applications (2016, methodology): “Besides its theoretical interest, the rank-one theorem has many applications in the theory of functions of bounded variation, we just mention the following: lowersemicontinuity and relaxation [9,18,26], integral representation theorems [13], Young measure theory [24, 25, 39], approximation theory [2”
Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints (2017, methodology): “This prevents, at the moment, the use of “pure” blow-up techniques and forces us to pass through the combination of the results of [10, 18] with the Young measure approach.”