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Applicable Analysis
An International Journal
Volume 103, 2024 - Issue 9
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Research Article

Γ-convergence of nonconvex unbounded integrals in strongly connected sets

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Pages 1704-1732 | Received 19 Apr 2023, Accepted 01 Sep 2023, Published online: 22 Sep 2023

References

  • Khruslov EY. The method of orthogonal projections and the Dirichlet boundary value problem in domains with a ‘fine-grained’ boundary. Mat Sb. 1972;17(1):38–60.
  • Khruslov EY. The first boundary value problem in domains with a complex boundary for higher order equations. Mat Sb. 1977;145(4):614–629.
  • Marchenko VA, Khruslov EY. Boundary value problems in domains with fine-grained boundary (in russian). Izdat. ‘Naukova Dumka’, Kiev, 1974.
  • Marchenko VA, Khruslov EY. Homogenization of partial differential equations. Vol. 46, Progress in Mathematical Physics. Boston, MA: Birkhäuser Boston, Inc.; 2006. Translated from the 2005 Russian original by M. Goncharenko and D. Shepelsky.
  • Cioranescu D, Jean Paulin JS. Homogenization in open sets with holes. J Math Anal Appl. 1979;71(2):590–607. doi: 10.1016/0022-247X(79)90211-7
  • Cioranescu D, Jean Paulin JS. Homogenization of reticulated structures. Vol. 136 Applied Mathematical Sciences. New York: Springer-Verlag; 1999.
  • Cioranescu D, Donato P. An introduction to homogenization. Vol. 17 Oxford Lecture Series in Mathematics and its Applications. New York: The Clarendon Press, Oxford University Press; 1999.
  • Chechkin GA, Piatnitski AL, Shamaev AS. Homogenization, Vol. 234 Translations of mathematical monographs. American Mathematical Society, Providence, RI, 2007. Methods and applications, Translated from the 2007 Russian original by Tamara Rozhkovskaya.
  • Jikov VV, Kozlov SM, Oleinik OA. Homogenization of differential operators and integral functionals. Berlin: Springer-Verlag; 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosifyan].
  • Braides A, Defranceschi A. Homogenization of multiple integrals. Vol. 12, Oxford Lecture Series in Mathematics and its Applications. New York: The Clarendon Press Oxford University Press; 1998.
  • Dal Maso G. An introduction to Γ-convergence: progress in nonlinear differential equations and their applications, Vol. 8, Boston (MA): Birkhäuser Boston Inc.; 1993.
  • Acerbi E, Chiadò Piat V, Dal Maso G, et al. An extension theorem from connected sets, and homogenization in general periodic domains. Nonlinear Anal. 1992;18(5):481–496. doi: 10.1016/0362-546X(92)90015-7
  • Braides A, Chiadò Piat V. Remarks on the homogenization of connected media. Nonlinear Anal. 1994;22(4):391–407. doi: 10.1016/0362-546X(94)90164-3
  • Kružík M, Roubíček T. Mathematical methods in continuum mechanics of solids. Interaction of Mechanics and Mathematics. Cham: Springer; 2019.
  • Ball JM. Some open problems in elasticity. In: Geometry, mechanics, and dynamics, New York: Springer; 2002. p. 3–59.
  • Anza Hafsa O, Clozeau N, Mandallena J-P. Homogenization of nonconvex unbounded singular integrals. Ann Math Blaise Pascal. 2017;24(2):135–193. doi: 10.5802/ambp.367
  • Anza Hafsa O, Leghmizi ML, Mandallena J-P. On a homogenization technique for singular integrals. Asymptot Anal. 2011;74(3–4):123–134.
  • Anza Hafsa O, Mandallena J-P. Homogenization of nonconvex integrals with convex growth. J Math Pures Appl. 2011;96(2):167–189. doi: 10.1016/j.matpur.2011.03.003
  • Anza Hafsa O, Mandallena J-P. Integral representation of unbounded variational functionals on Sobolev spaces. Ric Mat. 2023;72(1):193–234.
  • Anza Hafsa O, Mandallena J-P, Zorgati H. Homogenization of unbounded integrals with quasiconvex growth. Ann Mat Pura Appl. 2015;194(6):1619–1648. doi: 10.1007/s10231-014-0437-z
  • Anza Hafsa O. On the integral representation of relaxed functionals with convex bounded constraints. ESAIM Control Optim Calc Var. 2010;16(1):37–57. doi: 10.1051/cocv:2008063
  • Anza Hafsa O, Mandallena J-P. Radial representation of lower semicontinuous envelope. Boll Unione Mat Ital. 2014;7(1):1–18. doi: 10.1007/s40574-014-0001-1
  • Anza Hafsa O, Mandallena J-P. On the relaxation of unbounded multiple integrals. arXiv:1207.2652 2012.
  • Fonseca I. The lower quasiconvex envelope of the stored energy function for an elastic crystal. J Math Pures Appl. 1988;67(2):175–195.
  • Anza Hafsa O, Mandallena J-P. Relaxation of nonconvex unbounded integrals with general growth conditions in Cheeger-Sobolev spaces. Bull Sci Math. 2018;142:49–93. doi: 10.1016/j.bulsci.2017.09.002
  • Akcoglu MA, Krengel U. Ergodic theorems for superadditive processes. J Reine Angew Math. 1981;323:53–67.
  • Licht C, Michaille G. Global-local subadditive ergodic theorems and application to homogenization in elasticity. Ann Math Blaise Pascal. 2002;9(1):21–62. doi: 10.5802/ambp.149
  • Demengel F, Demengel G. Functional spaces for the theory of elliptic partial differential equations. Universitext. Springer, London; EDP Sciences, Les Ulis, 2012. Translated from the 2007 French original by Reinie Erné.

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