A proposal for the modification of the notion of quasiconvexity

This note proposes to modify the definition of quasiconvexity of a function f Úìm˜n r Ï ñ Ú ̈ ñ T −ðÙð( on the spaceìm˜n of m ˜ n matrices in such a way that (i) the polyconvexity implies quasiconvexity without any additional measurability or continuity assumption on f and (ii) the pointwise supremum of any family of quasiconvex functions is a quasiconvex function. Property (ii) allows one to define the quasiconvex envelope f qc of any f Ú ìm˜n r Ï ñ as the largest quasiconvex minorant of f ; this, in turn, makes it possible to establish the formula similar to that in Dacorogna [4, 6] for f qc: If E ⊂ ñn is a nonempty bounded open set with |ãE| ̈ 0 then for any A Xìm˜n we have f qc A ̈ inf "|E|−1 ˆ E f A + Du dx* where the infimum is taken over all u X W 1Ùð 0 EÙñm such that the integral ˆE f A+Du dx is well defined and there exists a partition of E into a set of measure 0 and a finite number of open sets such that Du is essentially constant on each of these open sets. The definition of quasiconvexity coincides with the original definition of Morrey [12], [13; Section 4.4] and Ball [1] if f is finite valued. If f Ú ìm˜n r −ðÙð then the definition coincides with those in [3, 10–11, 14]; if f has ð in its range then there are functions quasiconvex in the present sense but not quasiconvex in the sense of [3, 10–11, 14]. An example is given to show this. MSC 2000 49J45, 49J99