Abstract

Consider the functional If(u)=∫Ωf(u(x)) dxI_f(u)=\int_\Omega f(u(x))\, dx, where u=(u1,…,um)u=(u_1,\dots,u_m). Assume additionally that each uju_j is constant along WjW_j, some subspace of Rn{\bf R}^n. We find the family of cones Λ\Lambda in Rm{\bf R}^m such that every Λ\Lambda-convex function ff defines a functional IfI_f which is lower semicontinuous under the sequential weak ∗* convergence in L∞(Ω,Rm)L^\infty (\Omega,{\bf R}^m ). Then we apply our result to functionals acting on distributional kernels of differential operators. We also discuss the relations of our problem to the rank–one conjecture of Morrey.

Author information

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

Agnieszka Kałamajska

Institute of Mathematics of the Polish Academy of Science, ul. Śniadeckich 8, 00–950 Warszawa, Poland
and: Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa, Poland,

kalamajs@mimuw.edu.pl

Keywords

  • Lower semicontinuity
  • quasiconvexity
  • Young measures

Mathematics Subject Classification

49J45, 49J10, 35E10

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Suggested citation

A. Kałamajska. “On Lambda-Convexity Conditions in the Theory of Lower Semicontinuous Functionals.” Journal of Convex Analysis 10 (2003), No. 2, 419–436. https://doi.org/10.68381/jca1025

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Published by Heldermann Verlag, 2003. Rights now held by Banach Press.