Remarques sur les notions de polyconvexité, quasi-convexité et convexité de rang 1. (Remarks on polyconvexity, quasi-convexity and rank 1 convexity). (French) Zbl 0609.49007

In the paper under review functions \(f:{\mathbb{R}}^{nm}\to {\mathbb{R}}\) are considered. It is well-known that the sequential lower semicontinuity of the functional \[ F(u)=\int_{\Omega}f(Du)dx,\quad u\in W^{1,p}(\Omega;{\mathbb{R}}^ m) \] with respect to the weak topology of \(W^{1,p}(\Omega;{\mathbb{R}}^ m)\) is strictly related to the notion of quasiconvexity, that is \[ f(z)\leq \frac{1}{meas(Q)}\int_{Q}f(z+Du(x))dx,\quad z\in {\mathbb{R}}^{nm},\quad u\in C^ 1_ 0(Q;{\mathbb{R}}^ m) \] for one (hence for all) open bounded subset Q of \({\mathbb{R}}^ n.\)
Several useful notions in the calculus of variations are examined, such as convexity, quasiconvexity, polyconvexity, and rank 1 convexity. They are related by the implications \[ (1)\text{ convexity }\Rightarrow\text{ polyconvexity }\Rightarrow\text{ quasiconvexity }\Rightarrow\text{ rank 1 convexity} \] which in general cannot be reversed (the only open question regards the implication rank 1 convexity \(\Rightarrow\) quasiconvexity). Together with the notions above, the associated envelopes are considered: \[ f^{**}=\sup \{g\leq f: g\text{ convex}\};\quad Pf=\sup \{g\leq f: g\text{ polyconvex}\} \]
\[ Qf=\sup \{g\leq f: g\text{ quasiconvex}\};\quad Ef=\sup \{g\leq f: g\text{ rank 1 convex}\} \] and we have \[ (2)\quad f^{**}\leq Pf\leq Qf\leq Ef\leq f. \] The paper contains many examples of classes of functions for which some of the implications (1) become equivalences, or some of the inequalities (2) become equalities, and a lot of counterexamples are illustrated.
Reviewer: G.Buttazzo


49J45 Methods involving semicontinuity and convergence; relaxation
26B25 Convexity of real functions of several variables, generalizations
35J50 Variational methods for elliptic systems