An elementary proof of Marcellini Sbordone semicontinuity theorem

Tomáš G. Roskovec; Filip Soudský

Kybernetika (2023)

  • Volume: 59, Issue: 5, page 723-736
  • ISSN: 0023-5954

Abstract

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The weak lower semicontinuity of the functional F ( u ) = ∫ Ω f ( x , u , ∇ u ) d x is a classical topic that was studied thoroughly. It was shown that if the function f is continuous and convex in the last variable, the functional is sequentially weakly lower semicontinuous on W 1 , p ( Ω ) . However, the known proofs use advanced instruments of real and functional analysis. Our aim here is to present a proof understandable even for students familiar only with the elementary measure theory.

How to cite

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Roskovec, Tomáš G., and Soudský, Filip. "An elementary proof of Marcellini Sbordone semicontinuity theorem." Kybernetika 59.5 (2023): 723-736. <http://eudml.org/doc/299166>.

@article{Roskovec2023,
abstract = {The weak lower semicontinuity of the functional \[ F(u)=\int \_\{\Omega \}f(x,u,\nabla u) \{\rm d\} x \] is a classical topic that was studied thoroughly. It was shown that if the function $f$ is continuous and convex in the last variable, the functional is sequentially weakly lower semicontinuous on $W^\{1,p\}(\Omega )$. However, the known proofs use advanced instruments of real and functional analysis. Our aim here is to present a proof understandable even for students familiar only with the elementary measure theory.},
author = {Roskovec, Tomáš G., Soudský, Filip},
journal = {Kybernetika},
keywords = {convexity; sequential semicontinuity; calculus of variation; minimizer},
language = {eng},
number = {5},
pages = {723-736},
publisher = {Institute of Information Theory and Automation AS CR},
title = {An elementary proof of Marcellini Sbordone semicontinuity theorem},
url = {http://eudml.org/doc/299166},
volume = {59},
year = {2023},
}

TY - JOUR
AU - Roskovec, Tomáš G.
AU - Soudský, Filip
TI - An elementary proof of Marcellini Sbordone semicontinuity theorem
JO - Kybernetika
PY - 2023
PB - Institute of Information Theory and Automation AS CR
VL - 59
IS - 5
SP - 723
EP - 736
AB - The weak lower semicontinuity of the functional \[ F(u)=\int _{\Omega }f(x,u,\nabla u) {\rm d} x \] is a classical topic that was studied thoroughly. It was shown that if the function $f$ is continuous and convex in the last variable, the functional is sequentially weakly lower semicontinuous on $W^{1,p}(\Omega )$. However, the known proofs use advanced instruments of real and functional analysis. Our aim here is to present a proof understandable even for students familiar only with the elementary measure theory.
LA - eng
KW - convexity; sequential semicontinuity; calculus of variation; minimizer
UR - http://eudml.org/doc/299166
ER -

References

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  1. Acerbi, E., Fusco, N., , Arch. Rational Mech. Anal. 86 (1984), 2, 125-145. MR0751305DOI
  2. Alibert, J.-J., Dacorogna, B., , Arch. Rational Mech. Anal. 117 (1992), 2, 155-166. MR1145109DOI
  3. Ball, J. M., Kirchheim, B., Kristensen, J., , Calc. Var. Partial Differential Equations 11 (2000), 4, 333-359. MR1808126DOI
  4. Benešová, B., Kružík, M., , SIAM Rev. 59 (2017), 4, 703-766. MR3720354DOI
  5. Bourdin, B., Francfort, G. A., Marigo, J.-J., , J. Elasticity 91 (2008), 1-3, 5-148. MR2390547DOI
  6. Dacorogna, B., 10.1007/BFb0096144, Lecture Notes in Mathematics Vol. 922, Springer-Verlag, Berlin - New York 1982. MR0658130DOI10.1007/BFb0096144
  7. Dacorogna, B., , Springer Science and Business Media, 2007. MR2361288DOI
  8. Bois-Reymond, P. du, , Math. Ann. 15 (1879), 2, 283-314. MR1510012DOI
  9. Eisen, G., , Manuscripta Math. 27 (1979), 1, 73-79. MR0524978DOI
  10. Ekeland, I., 10.1016/0022-247X(74)90025-0, J. Math. Anal. Appl. 47 (1974), 324-353. Zbl0286.49015MR0346619DOI10.1016/0022-247X(74)90025-0
  11. Ekeland, I., , Bull. Amer. Math. Soc. (N.S.) 1 (1979), 3, 443-474. Zbl0441.49011MR0526967DOI
  12. Ekeland, I., Temam, R., Analyse convexe et problèmes variationnels., Collection Études Mathématiques. Dunod, Paris, Gauthier-Villars, Paris - Brussels - Montreal 1974. MR0463993
  13. Fonseca, I., Malý, J., 10.1016/s0294-1449(97)80139-4, Ann. Inst. H. Poincaré C Anal. Non Linéaire 14 (1997), 3, 309-338. MR1450951DOI10.1016/s0294-1449(97)80139-4
  14. Fonseca, I., Müller, S., , SIAM J. Math. Anal. 30 (1999), 6, 1355-1390. MR1718306DOI
  15. Giaquinta, M., Giusti, E., , Acta Math. 148 (1982), 31-46. MR0666107DOI
  16. Giusti, E., Direct Methods in the Calculus of Variations., World Scientific, 2003. MR1962933
  17. Grabovsky, Y., , Arch. Ration. Mech. Anal. 227 (2018), 2, 607-636. MR3740383DOI
  18. Guerra, A., Kristensen, J., , Arch. Ration. Mech. Anal. 245 (2022), 1, 479-500. MR4444078DOI
  19. Kałamajska, A., , Colloq. Math. 74 (1997), 1, 71-78. MR1455456DOI
  20. Kristensen, J., , Math. Ann. 313 (1999), 4, 653-710. MR1686943DOI
  21. Kristensen, J., , Nonlinear Anal. 120 (2015), 43-56. MR3348045DOI
  22. Kristensen, J., Rindler, F., , Arch. Ration. Mech. Anal. 197 (2010), 2, 539-598. MR2660519DOI
  23. Lagrange, J. L., Mécanique analytique, Vol. 1., Mallet - Bachelier, 1853. MR2858305
  24. Leoni, G., A First Course in Sobolev Spaces, Vol. 181 Graduate Studies in Mathematics. (Second edition.), American Mathematical Society, Providence 2017. MR3726909
  25. Lukeš, J., Malý, J., Measure and Integral. (Second edition.), Matfyzpress, Prague 2005. MR2316454
  26. Marcellini, P., 10.1007/BF01168345, Manuscr. Math. 51 (1985), 1-3, 1-28. MR0788671DOI10.1007/BF01168345
  27. Marcellini, P., Sbordone, C., , Nonlinear Anal. 4 (1980), 2, 241-257. MR0563807DOI
  28. Meyers, N. G., , Trans. Amer. Math. Soc. 119 (1965), 125-149. MR0188838DOI
  29. Mingione, G., , Appl. Math. 51 (2006), 4, 355-426. MR2291779DOI
  30. Morrey, Ch. B., Jr., , Pacific J. Math. 2 (1952), 25-53. MR0054865DOI
  31. Prinari, F., , NoDEA Nonlinear Differential Equations Appl. 22 (2015), 6, 1591-1605. MR3415015DOI
  32. Serrin, J., , Trans. Amer. Math. Soc. 101 (1961), 139-167. MR0138018DOI
  33. Sil, S., 10.1515/acv-2016-0058, Adv. Calc. Var. 12 (2019), 1, 57-84. MR3898186DOI10.1515/acv-2016-0058
  34. Tonelli, L., , Rendiconti del Circolo Matematico di Palermo (1884-1940), 44 (1920), 1, 167-249. DOI
  35. Verde, A., Zecca, G., , Nonlinear Anal. 71 (2009), 10, 4515-4524. MR2548683DOI
  36. Šverák, V., Rank-one convexity does not imply quasiconvexity., Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), 1-2, 185-189. MR1149994

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