Abstract and Applied Analysis

A New Extension of Serrin's Lower Semicontinuity Theorem

Xiaohong Hu and Shiqing Zhang

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We present a new extension of Serrin's lower semicontinuity theorem. We prove that the variational functional f x , u , u d x defined on W l o c 1,1 is lower semicontinuous with respect to the strong convergence in L l o c 1 , under the assumptions that the integrand f x , s , ξ has the locally absolute continuity about the variable x .

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Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 368610, 7 pages.

First available in Project Euclid: 26 February 2014

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Hu, Xiaohong; Zhang, Shiqing. A New Extension of Serrin's Lower Semicontinuity Theorem. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 368610, 7 pages. doi:10.1155/2013/368610. https://projecteuclid.org/euclid.aaa/1393444383

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  • R. A. Adams and J. F. Fournier, Sobolev Space, Academic Press, New York, NY, USA, 2nd edition, 2003.
  • C. Y. Pauc, La Méthode Métrique en Calcul des Variations, Hermann, Paris, France, 1941.
  • J. Serrin, “On the definition and properties of certain variational integrals,” Transactions of the American Mathematical Society, vol. 101, pp. 139–167, 1961.
  • L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Oxford University Press, New York, NY, USA, 2000.
  • V. de Cicco and G. Leoni, “A chain rule in ${L}^{1}(di{\lightv};\Omega )$ and its applications to lower semicontinuity,” Calculus of Variations and Partial Differential Equations, vol. 19, no. 1, pp. 23–51, 2004.
  • I. Fonseca and G. Leoni, “Some remarks on lower semicontinuity,” Indiana University Mathematics Journal, vol. 49, no. 2, pp. 617–635, 2000.
  • I. Fonseca and G. Leoni, “On lower semicontinuity and relaxation,” Proceedings of the Royal Society of Edinburgh A, vol. 131, no. 3, pp. 519–565, 2001.
  • M. Gori and P. Marcellini, “An extension of the Serrin's lower semicontinuity theorem,” Journal of Convex Analysis, vol. 9, no. 2, pp. 475–502, 2002.
  • M. Gori, F. Maggi, and P. Marcellini, “On some sharp conditions for lower semicontinuity in L1,” Differential and Integral Equations, vol. 16, no. 1, pp. 51–76, 2003.
  • E. de Giorgi, Teoremi di semicontinuit'a nel calcolo delle variazioni, Istituto Nazionale di Alta Matematica, Rome, Italy, 1968.