Polyconvexity equals rank-one convexity for connected isotropic sets in \(\mathbb M^{2\times 2}\). (English. Abridged French version) Zbl 1050.49010

Conti, Sergio; De Lellis, Camillo; Müller, Stefan; Romeo, Mario

C. R., Math., Acad. Sci. Paris 337, No. 4, 233-238 (2003).

For an integral functional of the form \[ I(u)= \int_\Omega W(\nabla u)\,dx, \] where \(u:\Omega\subset \mathbb{R}^n\to \mathbb{R}^m\), if \(K\) is the set of matrices where \(W\) attains its minimum, it is of interest to characterize the quasiconvex hull \(K^{qc}\) defined as the set of matrices \(A\) such that the relaxation \(\overline I\) of \(I\) attains its minimum on the linear function \(Ax\). Due to the difficulty to compute quasiconvex envelopes, other kinds of envelopes are considered, as the polyconvex hull \(K^{pc}\), the rank-one hull \(K^{rc}\), the lamination-convex hull \(K^{lc}\), suitably defined. Straightforward inclusions are: \[ K^{lc}\subset K^{rc}\subset K^{qc}\subset K^{pc}. \] In the paper, the particular case \(n= m= 2\) is considered. In this situation the authors prove that a compact connected \(\text{SO}(2)\) invariant set \(K\subset\mathbb{M}^{2\times 2}\) is lamination convex (i.e., \(K= K^{lc}\)) if and only if it is polyconvex (i.e., \(K= K^{pc}\)).
The same result, with some small additional assumptions, was first proved by Cardaliaguet and Tahraoui. In the paper under review the authors give a new shorter and self-contained proof, together with some examples which show that the connectedness assumption cannot be removed.

Reviewer: Giuseppe Buttazzo (Pisa)

MSC:

49J45	Methods involving semicontinuity and convergence; relaxation
52A40	Inequalities and extremum problems involving convexity in convex geometry

Keywords:

quasiconvexity; singular values; integral functionals

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