Does Rank-One Convexity Imply Quasiconvexity?

Abstract

Let Ω ⊂ R^m be a bounded open set. Let M^nxm denote the set of real n × m matrices and suppose that W: M^nxm → ֿR is Borel measurable and bounded below. (Here ֿR denotes the extended real line with its usual topology.) We are interested in the problem of minimizing

$${\text{I(u) = }}\int\limits_{\Omega } {\text{W}} {\text{(Du)(x))dx}}$$

((1.1))

among functions u ε W^1,1 (Ω;Rⁿ) satisfying appropriate boundary conditions. An important application is to nonlinear elasticity, when W = W(A) is the stored-energy function of a homogeneous material and u(x) is the deformed position of the particle at x ε Ω in a reference configuration; in this case we usually take m = n = 3, but the cases 1 ≤ m ≤ n ≤ 3 are also of interest and cover certain string and membrane problems. It is convenient to allow W to take the value + so as to include various constraints. In compressible nonlinear elasticity (m = n = 3), for example, we may set W(A) = +∞ for det A ≤ b, where b ≥ 0 is a constant, to reflect the fact that infinite energy is required to make a reflection of the body or to homogeneously compress it to b times its original volume. Similarly, for an incompressible material it is convenient to set W(A) = +∞ if and only if det A ≠ 1.