Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
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Γ-CONVERGENCE FOR AN OPTIMAL DESIGN PROBLEM WITH VARIABLE EXPONENT

  • HAMDI ZORGATI (DEPARTMENT OF MATHEMATICS AND STATISTICS, COLLEGE OF SCIENCES, IMAM MOHAMMAD IBN SAUD ISLAMIC UNIVERSITY (IMSIU))
  • Received : 2023.06.27
  • Published : 2023.12.25
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Abstract

In this paper, we derive the Γ-limit of functionals pertaining to some optimal material distribution problems that involve a variable exponent, as the exponent goes to infinity. In addition, we prove a relaxation result for supremal optimal design functionals with respect to the weak-∗ L∞(Ω; [0, 1])× W1,p0 (Ω;ℝm) weak topology.

Keywords

Acknowledgement

I wish to thank Professor Herve Le Dret for many useful discussions concerning the subject of this paper. The author's work was supported by IMSIU.

References

  1. H. ZORGATI: A Γ-convergence result for optimal design problems, C. R., Math., Acad. Sci. Paris., 360 (2022), 1145-1151. https://doi.org/10.5802/crmath.375
  2. E. ACERBI, G. MINGIONE, Regularity results for a class of functionals with non-standard growth, Arch. Rational Mech. Anal., 156 (2001), 121-140. https://doi.org/10.1007/s002050100117
  3. K. RAJAGOPAL, M. RUZICKA Mathematical modeling of electrorheological materials, Cont. Mech. and Thermodynamics, 13 (2001), 59-78. https://doi.org/10.1007/s001610100034
  4. M. RUZICKA Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin, 2000.
  5. A. WINEMAN, K. RAJAGOPAL On constitutive equations for electrorheological materials, Contin. Mech. Thermodyn., 7 (1995), 1-22. https://doi.org/10.1007/BF01175766
  6. S. ANTONTSEV, J. RODRIGUES, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara Sez. VII Sci. Mat., 52 (2006), 19-36. https://doi.org/10.1007/s11565-006-0002-9
  7. Y. CHEN, S. LEVINE, M. RAO, Variable Exponent, Linear Growth Functionals in Image Restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406. https://doi.org/10.1137/050624522
  8. F. LI, Z. LI, L. PI, Variable exponent functionals in image restoration, Applied Mathematics and Computation, 216 (2010), 870-882. https://doi.org/10.1016/j.amc.2010.01.094
  9. I. FONSECA, G. FRANCFORT, 3D-2D asymptotic analysis of an optimal design problem for thin films, J. Reine Angew. Math., 505 (1998), 173-202. https://doi.org/10.1515/crll.1998.505.173
  10. A. BRAIDES, I. FONSECA, G. FRANCFORT, 3D-2D asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J.,49 (2000), 1367-1404. https://doi.org/10.1512/iumj.2000.49.1822
  11. A. GARRONI, V. NESI, M. PONSIGLIONE, Dielectric breakdown: Optimal bounds, Proc. R. Soc. A,, 457 (2001) 2317-2335. https://doi.org/10.1098/rspa.2001.0803
  12. M. BOCEA, M. MIHILESCU, Γ-Convergence of power-law functionals with variable exponents, Nonlinear Analysis: Theory, Methods and Applications, 73 (2010), 110-121.
  13. M. BOCEA, V. NESI, Γ-convergence of power-law functionals, variational principles in L∞, and applications, SIAM J. Math. Anal., 39 (2008), 1550-1576. https://doi.org/10.1137/060672388
  14. N. ANSINI, F. PRINARI, Power-law approximation under differential constraints, SIAM J. Math. Anal., 46 (2014), 1085-1115. https://doi.org/10.1137/130911391
  15. N. ANSINI, F. PRINARI, On the lower semicontinuity of supremal functional under differential constraints, ESAIM, Control Optim. Calc. Var., 21 (2015), 1053-1075. https://doi.org/10.1051/cocv/2014058
  16. T. CHAMPION, L. DE PASCALE, F. PRINARI, Γ-convergence and absolute minimizers for supremal functionals, ESAIM: COCVC,, 10 (2004) 14-27. https://doi.org/10.1051/cocv:2003036
  17. M. ELEUTERI, F. PRINARI, Γ-convergence for power-law functionals with variable exponents, Nonlinear Analysis: Real World Applications, 58 (2021) 103221.
  18. G. DAL MASO, An Introduction to Γ-Convergence, Birkhauser, Boston, 1993.
  19. DE GIORGI E, Sulla convergenza di alcune successioni di integrali del tipo dell area, Rend. Mat., (IV), 8 (1975), 277-294.
  20. L. DIENING, P. HARJULEHTO, P. HASTO, M. RUZICKA, Lebesgue and Sobolev Spaces with Variable Exponents Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Berlin, 2011.
  21. I. SHARAPUDINOV, On the topology of the space Lp(t) ([0, 1]), Math. Notes., 26 (1979), 796-806. https://doi.org/10.1007/BF01159546
  22. C.B. MORREY JR, Quasiconvexity and the semicontinuity of multiple integrals, Pacific J. Math., 2 (1952), 25-53. https://doi.org/10.2140/pjm.1952.2.25
  23. H. LE DRET, A. RAOULT, Variational Convergence for Nonlinear Shell Models with Directors and Related Semicontinuity and Relaxation Results, Arch. Rational Mech. Anal., 154 (2000), 101-134. https://doi.org/10.1007/s002050000100
  24. I. FONSECA, S. MULLER, A-quasiconvexity, lower semicontinuity and Young measures, SIAM J. Math. Anal., 30 (1999), 1355-1390. https://doi.org/10.1137/S0036141098339885