Rank-one convex hulls in \(\mathbb R^{2\times 2}\). (English) Zbl 1104.49013

Székelyhidi, László jun.

Calc. Var. Partial Differ. Equ. 22, No. 3, 253-281 (2005).

A rank-one convex hull \(K^{\text{rc}}\) of a set of matrices \(K\subset{\mathbb R}^{m\times n}\) is defined standardly as \(\{A\in{\mathbb R}^{m\times n};\;\forall f:{\mathbb R}^{m\times n}\to{\mathbb R}\) rank-one convex: \(f(A)\leq\sup f(K)\}\) where \(f\) rank-one convex means that \(t\mapsto f(A+tB):\mathbb R\to\mathbb R\) is convex whenever rank\(\,B=1\).
In the paper, for \(m=2=n\), it is shown that \(K^{\text{rc}}=K\) if \(K\) does not contain two matrices whose difference has the rank one and if \(K\) does not contain any quadruple of matrices forming a so-called (Tartar’s) \(T_4\) configuration (i.e. a special hierarchical family of matrices with no rank-one connections).
In particular, the paper shows that any compact \(K\subset\mathbb R^{2\times2}\), which does not contain two matrices whose difference has the rank equal to one and for which \(K^{\text{rc}}\neq K\), contains a \(T_4\) configuration. Furthermore, a simple numerical criterion for testing \(T_4\) configurations is given.

Reviewer: Tomáš Roubíček (Praha)

MSC:

49J45	Methods involving semicontinuity and convergence; relaxation
52A30	Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
26B25	Convexity of real functions of several variables, generalizations

Keywords:

rank-one convexity

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