Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Pablo Pedregal

Dep. de Matemáticas, ETSI Industriales, Universidad de Castilla - La Mancha, 13071 Ciudad Real, Spain

pablo.pedregal@uclm.es

Keywords

  • Inner variations
  • multiplicative quasiconvexity
  • adaptivity

Mathematics Subject Classification

37C40, 37E05, 49J45

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Suggested citation

P. Pedregal. “Some Remarks on Quasiconvexity, Inner Variations, and Optimal Meshes.” Journal of Convex Analysis 15 (2008), No. 2, 381–393. https://doi.org/10.68381/jca15027

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Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.