Journal of Convex Analysis 08 (2001), No. 2, 369--386
Copyright Heldermann Verlag 2001
Quasi Convex Integrands and Lower Semicontinuity in BV
Primo Brandi
Dept. of Mathematics and Informatics, University of Perugia, Via L. Vanvitelli 1, 06123 Perugia, Italy
Anna Salvadori
Dept. of Mathematics and Informatics, University of Perugia, Via L. Vanvitelli 1, 06123 Perugia, Italy
We discuss the lower semicontinuity of multiple integrals of the calculus of variations, for quasi convex integrands in BV setting. The main result is a lower semicontinuity theorem with respect to weak convergence in BV, i.e. L1-convergence of equi-BV sequences, under mild assumptions on the integrand. The approach we propose here is based on two main results:
(1) the characterization of the lower semicontinuity of a generic sequence of integrals in terms of a local condition, called lower mean-value convergence (l-mv);
(2) the asymptotic behavior of the essential gradients of weak convergent sequences in BV, expressed by means of a local convergence condition, called mean-value condition (mv).
By virtue of (mv)-convergence of the gradients, in the case of multiple integrals of the calculus of variations, (l-mv)-condition reduces to a suitable localization of a sequential Jensen's-type inequality (Js). A deep analysis of weak convergent sequences in BV allows us to get the key result of the paper which links up the local oscillation of the surfaces with (mv)-convergence of the gradients and with (Js)-inequality on the integrals.
For the sake of comparison with the literature on the subject, we wish to mention that our main semicontinuity theorem can be considered as an extension to BV-setting of the result due to Fonseca-Muller ["Quasi-convex integrands and lower semicontinuity in L1", SIAM J. Math. Anal. 23 (1992) 1081--1098], with an improvement of the assumptions on the integrand. Moreover, it improves the semicontinuity theorem we proved in the particular case of convex integrands. Finally, we wish to mention that the present research finds applications to the study of a variational model for the plastic deformation of beams and plates under loads of different types.
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