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Revista Matemática Complutense

, Volume 32, Issue 2, pp 327–351 | Cite as

Variational problems with variable regular bilateral constraints in variable domains

  • Alexander A. KovalevskyEmail author
Article
  • 93 Downloads

Abstract

We establish conditions for the convergence of minimizers and minimum values of integral and more general functionals on sets of functions defined by bilateral constraints in variable domains. The constraints and domains under consideration depend on the same natural parameter, and the constraints belong to Sobolev spaces related to the given domains. The main condition on the constraints is the convergence to zero of the measure of the set where the difference between the upper and lower constraints is less than a positive measurable function.

Keywords

Integral functional Variational problem Bilateral constraints Minimizer Minimum value \({\Gamma }\)-convergence Variable domains 

Mathematics Subject Classification

49J40 49J45 

Notes

Acknowledgements

This work was partially supported by the Russian Academic Excellence Project (Agreement No. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

References

  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Amaziane, B., Goncharenko, M., Pankratov, L.: \(\varGamma _D\)-convergence for a class of quasilinear elliptic equations in thin structures. Math. Methods Appl. Sci. 28(15), 1847–1865 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization, 2nd edn. SIAM, Philadelphia (2014)CrossRefzbMATHGoogle Scholar
  4. 4.
    Attouch, H., Picard, C.: Variational inequalities with varying obstacles: the general form of the limit problem. J. Funct. Anal. 50(3), 329–386 (1983)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dal Maso, G.: Asymptotic behaviour of minimum problems with bilateral obstacles. Ann. Mat. Pura Appl. 129(1), 327–366 (1981)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dal Maso, G.: \(\varGamma \)-convergence and \(\mu \)-capacities. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 14(3), 423–464 (1987)Google Scholar
  7. 7.
    Dal Maso, G.: An Introduction to \({\varGamma }\)-Convergence. Birkhäuser, Boston (1993)CrossRefGoogle Scholar
  8. 8.
    Dal Maso, G., Longo, P.: \(\varGamma \)-limits of obstacles. Ann. Mat. Pura Appl. 128(1), 1–50 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    De Giorgi, E., Franzoni, T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. (8) 58(6), 842–850 (1975)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Khruslov, E.Ya.: The asymptotic behavior of solutions of the second boundary value problem under fragmentation of the boundary of the domain. Math. USSR-Sb. 35(2), 266–282 (1979)Google Scholar
  11. 11.
    Kovalevskii, A.A.: Some problems connected with the problem of averaging variational problems for functionals with a variable domain. In: Mitropol’skii, Yu. A. (ed.) Current Analysis and its Applications, pp. 62–70. Naukova Dumka, Kiev (1989) [in Russian]Google Scholar
  12. 12.
    Kovalevskii, A.A.: Conditions for \({\varGamma }\)-convergence and homogenization of integral functionals with different domains. Dokl. Akad. Nauk Ukrain. SSR No. 4, 5–8 (1991). [in Russian]MathSciNetGoogle Scholar
  13. 13.
    Kovalevskii, A.A.: On necessary and sufficient conditions for the \({\varGamma }\)-convergence of integral functionals with different domains of definition. Nelinein. Granichnye Zadachi 4, 29–39 (1992). [in Russian]Google Scholar
  14. 14.
    Kovalevskii, A.A.: \(G\)-convergence and homogenization of nonlinear elliptic operators in divergence form with variable domain. Russ. Acad. Sci. Izv. Math. 44(3), 431–460 (1995)Google Scholar
  15. 15.
    Kovalevskii, A.A.: On the \(\varGamma \)-convergence of integral functionals defined on Sobolev weakly connected spaces. Ukr. Math. J. 48(5), 683–698 (1996)CrossRefGoogle Scholar
  16. 16.
    Kovalevsky, A.A.: On \(L^1\)-functions with a very singular behaviour. Nonlinear Anal. 85, 66–77 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kovalevsky, A.A.: On the convergence of solutions to bilateral problems with the zero lower constraint and an arbitrary upper constraint in variable domains. Nonlinear Anal. 147, 63–79 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kovalevsky, A.A.: On the convergence of solutions of variational problems with bilateral obstacles in variable domains. Proc. Steklov Inst. Math. 296(suppl. 1), 151–163 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Malý, J., Ziemer, W.P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations. AMS, Providence (1997)CrossRefzbMATHGoogle Scholar
  20. 20.
    Marchenko, V.A., Khruslov, E.Ya.: Homogenization of Partial Differential Equations. Birkhäuser, Boston (2006)Google Scholar
  21. 21.
    Murat, F.: Sur l’homogeneisation d’inequations elliptiques du 2ème ordre, relatives au convexe \(K(\psi _1,\psi _2)=\{v\in H^1_0(\varOmega )\, | \,\psi _1\leqslant v\leqslant \psi _2\, {\rm p. p.}\, {\rm dans}\, \varOmega \}\). Publ. Laboratoire d’Analyse Numérique, No. 76013, Univ. Paris VI (1976)Google Scholar
  22. 22.
    Pankratov, L.: \(\varGamma \)-convergence of nonlinear functionals in thin reticulated structures. C. R. Math. Acad. Sci. Paris 335(3), 315–320 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rudakova, O.A.: On \({\varGamma }\)-convergence of integral functionals defined on various weighted Sobolev spaces. Ukr. Math. J. 61(1), 121–139 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Vainberg, M.M.: Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations. Wiley, New York (1974)Google Scholar
  25. 25.
    Velichkov, B.: Existence and Regularity Results for Some Shape Optimization Problems. Tesi. Scuola Normale Superiore, Pisa (2015)CrossRefzbMATHGoogle Scholar
  26. 26.
    Zhikov, V.V.: Questions of convergence, duality, and averaging for functionals of the calculus of variations. Math. USSR-Izv. 23(2), 243–276 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zhikov, V.V.: On passage to the limit in nonlinear variational problems. Russ. Acad. Sci. Sb. Math. 76(2), 427–459 (1993)MathSciNetzbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and MechanicsUral Branch of the Russian Academy of SciencesYekaterinburgRussia
  2. 2.Institute of Natural Sciences and MathematicsUral Federal UniversityYekaterinburgRussia

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