Journal of Convex Analysis 15 (2008), No. 2, 381--393
Copyright Heldermann Verlag 2008
Some Remarks on Quasiconvexity, Inner Variations, and Optimal Meshes
Pablo Pedregal
Dep. de Matemáticas, ETSI Industriales, Universidad de Castilla - La Mancha, 13071 Ciudad Real, Spain
pablo.pedregal@uclm.es
We explore a formulation of quasiconvexity in terms of inner variations instead of variations of the dependent variable. This leads to a specific transformation of integrands that is stable for rank-one convexity, quasiconvexity, and polyconvexity, but not for convexity. An interesting application of these ideas is concerned with the analysis of optimal adaptive meshes for variational problems. This theme is not new either in the analytical or numerical treatment. We complete the discussion with some easy examples in dimension one, and defer the much more complex situation in higher dimension for a later work. An interesting point is that our remarks are valid when the number of components for fields is not greater than the number of independent variables.
Keywords: Weak convergence, Young measure, invariant measure.
MSC: 37C40, 37E05, 49J45
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