Abstract
We study the L 1-lower semicontinuity in BV of an integral functional of the type \({{\displaystyle{{ \int_{{\Omega}}f(x,u,\nabla u)dx+ \int_{{\Omega}} {{-\kern-,375cm\int}}_{{u^-(x)}}^{{u^+(x)}}f^\infty(x,t,\frac{{D^su}}{{|D^su|}})dt d|D^su|}}}}\). Our assumptions on f extend previous results recently obtained by Gori, Maggi and Marcellini in the case where the above functional is restricted to W 1,1.
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References
Ambrosio, L.: New lower semicontinuity results for inegral functionals. Rend. Accad. Naz. Sci. XL 1, 1–42 (1987)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford University Press, 2000
De Cicco, V.: A lower semicontinuity result for functionals defined on BV(Ω). Ricerche di Matematica 39, 293–325 (1990)
De Cicco, V., Leoni, G.: A chain rule in L 1(div;Ω) and its applications to lower semicontinuity. Preprint, 2002
De Giorgi, E.: Teoremi di semicontinuità nel calcolo delle variazioni. Istituto Nazionale di Alta Matematica, 1968-1969
De Giorgi, E., Buttazzo, G., Dal Maso, G.: On the lower semicontinuity of certain integral functions. Atti Accad. Naz. Lincei, Cl. Sc. Fis. Mat. Natur. Rend. 74, 274–282 (1983)
Fonseca, I., Leoni, G.: Some remarks on lower semicontinuity. Indiana Univ. Math. J. 49, 617–635 (2000)
Fonseca, I., Leoni, G.: On lower semicontinuity and relaxation. Proc. R. Soc. Edim., Sect. A, Math. 131, 519–565 (2001)
Gori, M., Marcellini. P.: An extension of the Serrin’s lower semicontinuity theorem. To appear on J. Convex Anal.
Gori, M., Maggi, F., Marcellini, P.: Some sharp conditions for lower semicontinuity in L 1. Preprint, 2002
Serrin, J.: On the definition and properties of certain variational integrals. Trans. Am. Math. Soc. 161, 139–167 (1961)
Vol’pert, A.I.: Spaces BV and quasi-linear equations. Math. USSR Sb. 17, 225–267 (1967)
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Fusco, N., Giannetti, F. & Verde, A. A remark on the L 1-lower semicontinuity for integral functionals in BV. manuscripta math. 112, 313–323 (2003). https://doi.org/10.1007/s00229-003-0400-6
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DOI: https://doi.org/10.1007/s00229-003-0400-6