Advertisement

Annali di Matematica Pura ed Applicata

, Volume 117, Issue 1, pp 139–152

# Periodic solutions and homogenization of non linear variational problems

• Paolo Marcellini
Article

## Summary

We prove an homogenization formula for some non linear variational problems that extends the analogous one known in the linear case. Namely the solution uε of the problem
$$\int\limits_\Omega {\left\{ {f\left( {\frac{x}{\varepsilon },Du_\varepsilon } \right) - \varphi \left( x \right)u_\varepsilon } \right\}dx} = minimum,$$
where the boundary data are independent of ɛ and f=f(x, ξ) is periodic in x, converges in some Lp space as ɛ goes to zero to the solution of an analogous variational problem whose integrand can be evaluated from f.

## Keywords

Periodic Solution Variational Problem Boundary Data Linear Case Homogenization Formula
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
A. AmbrosettiC. Sbordone,Γ convergenza e G-convergenza per problemi non lineari di tipo ellittico, Boll. Un. Mat. Ital.,13-A (1976), pp. 352–362.
2. [2]
I. Babuška,Solution of interface problems by homogenization I, II, III, Techn. Note, Univ. of Maryland (1974–75).Google Scholar
3. [3]
I. Babuška,Homogenization and its application. Mathematical and computational problems, Proc. Symp. Numerical Sol. Partial Diff. Eq., III, Maryland (1975), Acad. Press, 1976, pp. 89–116.Google Scholar
4. [4]
N. S. Bakhvalov,Averaging of partial differential equations with rapidly oscillating coefficients, Dokl. Akad. Nauk SSSR,221 (1975), pp. 516–519; Soviet Math. Dokl.,16 (1975), pp. 351–355.
5. [5]
A. BensoussanJ. L. LionsG. Papanicolaou,Sur quelques phénomènes asymptotiques stationnaires, C. R. Acad. Sc. Paris,281 (1975), pp. 89–94.
6. [6]
A. BensoussanJ. L. LionsG. Papanicolaou,Sur quelques phénomènes asymptotiques d'évolution, C. R. Acad. Sc. Paris,281 (1975), pp. 317–322.
7. [7]
A. BensoussanJ. L. LionsG. Papanicolaou,Sur de nouveaux problèmes asymptotiques, C. R. Acad. Sc. Paris,282 (1976), pp. 143–147.
8. [8]
A. BensoussanJ. L. LionsG. Papanicolaou,Homogénéisation, correcteurs et problèmes non-linéaires, C. R. Acad. Sc. Paris,282 (1976), pp. 1277–1282.
9. [9]
L. BoccardoI. Capuzzo Dolcetta,G-convergenza e problema di Dirichlet unilaterale, Boll. Un. Mat. Ital.,12 (1975), pp. 115–123.
10. [10]
L. BoccardoP. Marcellini,Sulla convergenza delle soluzioni di disequazioni variazionali, Ann. Mat. Pura Appl.,110 (1976), pp. 137–159.
11. [11]
O. Caligaris,Sulla convergenza delle soluzioni di problemi di evoluzione associati a sottodifferenziali di funzioni conversse, Boll. Un. Mat. Ital.,13-B (1976), pp. 778–806.
12. [12]
F. ColombiniS. Spagnolo,Sur la convergence de solutions d'équations paraboliques avec des coefficients qui dépendent du temps, C. R. Acad. Sc. Paris,282 (1976), pp. 735–737.
13. [13]
E. De GiorgiS. Spagnolo,Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital.,8 (1973), pp. 391–411.
14. [14]
J. L. Lions,Asymptotic behaviour of solutions of variational inequalities with highly oscillating coefficients, Lect. Notes in Math., Springer,503 (1976), pp. 30–55.
15. [15]
J. L. Lions,Sur quelques questions d'analyse, de mécanique et de contrôle optimal, Univ. de Montréal (1976).Google Scholar
16. [16]
P. Marcellini,Su una convergenza di funzioni convesse, Boll. Un. Mat. Ital.,8 (1973), pp. 137–158.
17. [17]
P. Marcellini — C. Sbordone,Homogenization of non-uniformly elliptic operators, Applicable Anal., to appear.Google Scholar
18. [18]
P. MarcelliniC. Sbordone,Sur quelques questions de G-convergence et d'homogénéisation non linéaire, C. R. Acad. Sc. Paris,284 (1977), pp. 535–537.
19. [19]
F. Murat,Sur l'homogénéisation d'inéquations elliptiques du 2ème ordre, relatives au convexe K(ω 1, ω2)={ v ∈ H 01 (Ω)|ω 1⩽v⩽ω2 p. p. dans Ω}, Univ. Paris VI, november 1976.Google Scholar
20. [20]
E. Sanchez Palencia,Solutions périodiques par rapport aux variables d'espace et applications, C. R. Acad. Sc. Paris,271 (1970), pp. 1129–1132.
21. [21]
E. Sanchez Palencia,Comportement local et macroscopique d'un type de milieux physiques hétérogènes, Internat. J. Engrg. Sci.,12 (1974), pp. 331–351.
22. [22]
C. Sbordone,Sulla G-convergenza di equazioni ellittiche e paraboliche, Ricerche Mat.,24 (1975), pp. 76–136.
23. [23]
C. Sbordone,Su alcune applicazioni di un tipo di convergenza variazionale, Ann. Scuola Norm. Sup. Pisa,2 (1975), pp. 617–638.
24. [24]
S. Spagnolo,Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Ann. Scuola Norm. Sup. Pisa,22 (1968), pp. 571–597.
25. [25]
S. Spagnolo,Convergence in energy for elliptic operators, Proc. Symp. Numerical Sol. Partial Diff. Eq., III, Maryland (1975), Acad. Press, 1976, pp. 469–498.Google Scholar
26. [26]
L. Tartar,Quelques remarques sur l'homogénéisation, Colloque France-Japan, 1976.Google Scholar

## Copyright information

© Nicola Zanichelli Editore 1978

## Authors and Affiliations

• Paolo Marcellini
• 1
1. 1.Firenze