Non-Markovian quadratic forms obtained by homogenization

Marc Briane

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 2, page 323-337
  • ISSN: 0392-4041

Abstract

top
This paper is devoted to the asymptotic behaviour of quadratic forms defined on L 2 . More precisely we consider the Γ -convergence of these functionals for the L 2 -weak topology. We give an example in which some limit forms are not Markovian and hence the Beurling-Deny representation formula does not hold. This example is obtained by the homogenization of a stratified medium composed of insulating thin-layers.

How to cite

top

Briane, Marc. "Non-Markovian quadratic forms obtained by homogenization." Bollettino dell'Unione Matematica Italiana 6-B.2 (2003): 323-337. <http://eudml.org/doc/195346>.

@article{Briane2003,
abstract = {This paper is devoted to the asymptotic behaviour of quadratic forms defined on $L^\{2\}$. More precisely we consider the $\Gamma$-convergence of these functionals for the $L^\{2\}$-weak topology. We give an example in which some limit forms are not Markovian and hence the Beurling-Deny representation formula does not hold. This example is obtained by the homogenization of a stratified medium composed of insulating thin-layers.},
author = {Briane, Marc},
journal = {Bollettino dell'Unione Matematica Italiana},
keywords = {non-equicoercive quadratic form; homogenization; stratified structure},
language = {eng},
month = {6},
number = {2},
pages = {323-337},
publisher = {Unione Matematica Italiana},
title = {Non-Markovian quadratic forms obtained by homogenization},
url = {http://eudml.org/doc/195346},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Briane, Marc
TI - Non-Markovian quadratic forms obtained by homogenization
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/6//
PB - Unione Matematica Italiana
VL - 6-B
IS - 2
SP - 323
EP - 337
AB - This paper is devoted to the asymptotic behaviour of quadratic forms defined on $L^{2}$. More precisely we consider the $\Gamma$-convergence of these functionals for the $L^{2}$-weak topology. We give an example in which some limit forms are not Markovian and hence the Beurling-Deny representation formula does not hold. This example is obtained by the homogenization of a stratified medium composed of insulating thin-layers.
LA - eng
KW - non-equicoercive quadratic form; homogenization; stratified structure
UR - http://eudml.org/doc/195346
ER -

References

top
  1. ARBOGAST, T.- DOUGLAS, J.- HORNUNG, U., Derivation of the double porosity model of single phase flow via homogenization theory, S.I.A.M. J. Math. Anal., 21 (1990), 823-836. Zbl0698.76106MR1052874
  2. BELLIEUD, M.- BOUCHITTÉ, G., Homogenization of elliptic problems in a fiber reinforced structure. Nonlocal effects, Annali della Scuola Normale Superiore di Pisa, 26, (4) (1998), 407-436. Zbl0919.35014MR1635769
  3. BELLIEUD, M.- BOUCHITTÉ, G., Homogenization of degenerate elliptic equations in a fiber structure, preprint 98/09 ANLA, Univ. Toulon. 
  4. BEURLING, A.- DENY, J., Espaces de Dirichlet, Acta Matematica, 99 (1958), 203-224. Zbl0089.08106MR98924
  5. BEURLING, A.- DENY, J., Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 208-215. Zbl0089.08201MR106365
  6. BRIANE, M., Homogenization in some weakly connected domains, Ricerche di Matematica, XLVII, no. 1 (1998), 51-94. Zbl0928.35037MR1760323
  7. DAL MASO, G., An introduction to Γ -convergence, Birkhäuser, Boston (1993). Zbl0816.49001MR1201152
  8. FENCHENKO, V. N.- KHRUSLOV, E. YA., Asymptotic of solution of differential equations with strongly oscillating and degenerating matrix of coefficients, Dokl. AN Ukr. SSR, 4 (1980). Zbl0426.35016
  9. FUKUSHIMA, M., Dirichlet Forms and Markov Processes, North-Holland Math. Library, 23, North-Holland and Kodansha, Amsterdam (1980). Zbl0422.31007MR569058
  10. KHRUSLOV, E. YA., The asymptotic behavior of solutions of the second boundary value problems under fragmentation of the boundary of the domain, Maths. USSR Sbornik, 35, no. 2 (1979). Zbl0421.35019
  11. KHRUSLOV, E. YA., Homogenized models of composite media, Composite Media and Homogenization Theory, G. Dal Maso and G. F. Dell'Antonio editors, in Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser (1991), 159-182. Zbl0737.73009MR1145750
  12. LE JEAN, Y., Mesures associées à une forme de Dirichlet. Applications, Bull. Soc. Math. de France, 106 (1978), 61-112. Zbl0393.31008
  13. MOSCO, U., Composite media and asymptotic Dirichlet forms, Journal of Functional Analysis, 123, no. 2 (1994), 368-421. Zbl0808.46042MR1283033

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.