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

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.

### 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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### References:

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