BURKHOLDER INTEGRALS, MORREY'S PROBLEMAND QUASICONFORMAL MAPPINGS

摘要

A stands for an arbitrary linear mapping (or its matrix) and Q C is any bounded domain. In other words. one requires that compactly supported perturbations of linear maps do not decrease the value of the integral. This notion is of fundamental importance in the calculus of variations as it is known to characterize lower semicontinuous integrals [37]. A weaker notion is that of rank-one convexity, which requires just that t E(A tX) is convex for any fixed matrix A and for any rank one matrix X. Rank-one convexity of an integrand is a local condition and thus much easier to verify than quasiconvexity. That quasiconvexity implies rank-one convexity was known after Money's fundamental work in the 1950s, but one had to wait until Sverak's paper [46] to find out that the converse is not true.However. Sverak's example works only in dimensions n 3. [43]. This leaves the possibility for a different outcome in dimension 2: see [26]. [40] for evidence in this direction. Money himself was not quite definite in which direction he thought things should be true; see [37]. [381. and [11, Sect. 9]. We reveal our own thoughts on the- matter by recalling the following conjecture in the spirit of Money: