On conditions for polyconvexity

Abstract

We give an example of a smooth function f: R2x2 2R, which is not polyconvex and which has the property that its restriction to any ball B C JR2x2 of radius one can be extended to a smooth polyconvex function fB :R 2X2 --+ R. In particular, it implies that there exists no 'local condition' which is necessary and sufficient for polyconvexity of functions g: RnX m -I+, where n, m > 2. We also briefly discuss connections with quasiconvexity. The class of polyconvex functions was introduced by Ball in [3] following earlier work by Morrey [9]. We recall that a function f: Rnxm --1 R is polyconvex if f(X) can be written as a convex function of the minors of X. For example, if m = n = 2, then f is polyconvex if there exists a convex function F : R 2X2 x R1 -]1R such that f(X) = F(X, detX) (detX denotes the determinant of the matrix X). The main reason for the interest in polyconvexity stems from the fact that, at present, it is the ouly known tractable condition which is reasonably flexible (e.g., it is compatible with natural requirements for stored energy functions in non-linear elasticity) and which implies quasiconvexity. The notion of quasiconvexity is due to Morrey [9] and is the natural substitute for convexity in the multi-dimensional calculus of variations. The condition of polyconvexity is strictly stronger than the condition of quasiconvexity (see, e.g., [1]). We refer to the monograph of Dacorogna [4] and the lecture notes by Miller [10] for the definitions and an introduction to the various convexity notions studied in the multi-dimensional calculus of variations. The principal result of this paper is contained in the following theorem. Theorem 1. Assume that n, m > 2. There exists a non-polyconvex smooth function f: RnX , such that its restriction to any ball B C R nxm of radius one can be extended to a smooth polyconvex function fB: RJnX, -R2. If n > 3, m > 2, then f can be taken to be non-quasiconvex. Remarks. (1) For m > 2, n > 6 Sverak (see [10]) has given an example of a nonquasiconvex smooth function f: R'xm R, which has the property that it agrees with polyconvex functions on balls of radius 1. (2) The result stated in Theorem 1 was announced in [5]. As an immediate consequence we deduce that there exists no 'local condition', which is equivalent to polyconvexity. To formulate this statement precisely we Received by the editors July 29, 1998. 1991 Mathematics Subject Classification. Primary 49J10, 49J45.