According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\textrm {GL}^{\!+}(2)$ of invertible $2\times 2$ - - matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander Mielke, it has been conjectured that the equivalence of rank-one convexity and polyconvexity holds for isotropic functions on $\textrm {GL}^{\!+}(2)$ as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function $W\colon \textrm {GL}^{\!+}(2)\to \mathbb {R}$ which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.

In this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued function W; SL(2) → ℝ with W(RF) = W(FR) = W(F) for all F ∈ SL(2) and all R ∈ SO(2), where SL(2) and SO(2) denote the special linear group and the special orthogonal group, respectively. In order to obtain our result, we combine earlier work by Dacorogna and Koshigoe on the relaxation of certain conformal planar energy functions with a recent result on the equivalence between polyconvexity and rank-one convexity for objective and isotropic energies in planar incompressible nonlinear elasticity.

We show that, in the two-dimensional case, every objective, isotropic and isochoric energy function that is rank-one convex on GL+(2) is already polyconvex on GL+(2). Thus, we answer in the negative Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasi-convexity. Our methods are based on different representation formulae for objective and isotropic functions in general, as well as for isochoric functions in particular. We also state criteria for these convexity conditions in terms of the deviatoric part of the logarithmic strain tensor.

Continuing earlier work by Székelyhidi, we describe the topological and geometric structure of so-called T 4-configurations which are the most prominent examples of nontrivial rank-one convex hulls. It turns out that the structure of T 4-configurations in $\mathbb{R}^{2\times 2}$ is very rich; in particular, their collection is open as a subset of $(\mathbb{R}^{2\times2})^{4}$ . Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect T 4-configurations.

We consider the lower semicontinuous functional of the form $I_f(u)=\int_\Omega f(u){\rm d}x$ where u satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar's Λ-convexity condition for the integrand f extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply to quasiconvex functions.

In this paper, we consider a Borel measurable function on the space of $\scriptstyle m\times n$ matrices $\scriptstyle f: M^{m\times n}\rightarrow \bar{\mathbb{R}}$ taking the value $ \scriptstyle +\infty$, such that its rank-one-convex envelope $\scriptstyle Rf$ is finite and satisfies for some fixed $\scriptstyle p>1$: $$\scriptstyle -c_0\leq Rf(F)\leq c(1+\Vert F\Vert^p)\ \hbox{for all}\ F\in M^{m\times n},$$ where $\scriptstyle c,c_0>0$. Let $\scriptstyle\O$ be a given regular bounded open domain of $\scriptstyle \mathbb{R}^n$. We define on $\scriptstyle W^{1,p}(\O;\mathbb{R}^m)$ the functional $$\scriptstyle I(u)=\int_{\O}f(\nabla u(x))\ dx.$$ Then, under some technical restrictions on $\scriptstyle f$, we show that the relaxed functional $\scriptstyle\bar I$ for the weak topology of $\scriptstyle W^{1,p}(\O;\mathbb{R}^m)$ has the integral representation: $$\scriptstyle \bar I(u)=\int_{\O}Q[Rf](\nabla u(x))\ dx,$$ where for a given function $\scriptstyle g$, $\scriptstyle Qg$ denotes its quasiconvex envelope.