×

Asymptotic behaviour of Dirichlet problems with measure data in perforated domains. (English) Zbl 0861.35009

The Dirichlet problem in a perforated domain \[ Lv_h=\nu,\quad x\in\Omega_h\subset\mathbb{R}^n;\quad v_h=0,\quad x\in \partial\Omega_h \] is studied. Here, \(L\) is a linear elliptic operator of the second order in divergence form in \(\Omega\) and \(\Omega_h\) is an arbitrary sequence of open subsets of \(\Omega\), the right hand side \(\nu\) is a measure with bounded variation. The convergence of a subsequence \(v_{hk}\) to the solution \(v\) of the problem \[ Lv+\mu v=\nu,\quad x\in\Omega;\quad v=0,\quad x\in\partial\Omega \] is proved under the assumption that \(\nu\) does not charge polar sets. It is supposed that in the case \(\nu\in H^{-1}(\Omega)\) this result takes place and a Borel measure \(\mu\) not charging polar sets is taken the same.
Reviewer: L.Kalyakin (Ufa)

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J25 Boundary value problems for second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1002/cpa.3160350206 · Zbl 0459.60069 · doi:10.1002/cpa.3160350206
[2] Ash, R.B. ”Real analysis and probability”. New York · Zbl 0206.18001
[3] Baxter J., Trans. Amer. Math. Soc., 303 pp 1– (1987)
[4] DOI: 10.1007/BF01442391 · Zbl 0762.49017 · doi:10.1007/BF01442391
[5] Cioranescu D., Pitman, London 70 pp 154– (1983)
[6] DOI: 10.1142/S0218202594000224 · Zbl 0804.47050 · doi:10.1142/S0218202594000224
[7] DOI: 10.1007/BF00276841 · Zbl 0634.35033 · doi:10.1007/BF00276841
[8] DOI: 10.1007/BF01442645 · Zbl 0644.35033 · doi:10.1007/BF01442645
[9] Dal Maso G., Houston J. Math., 15 pp 35– (1989)
[10] Finzi Vita S., Asymptotic 5 pp 269– (1992)
[11] Gilbarg D., Springer-Verlag
[12] Kacimi H., In Partial Differential equations and the Calculus of Variations, Essays in Honor of Ennio De Giorgi
[13] Malusa A., Ann. Mat. Pura Appl., to appear. 5 (1992)
[14] DOI: 10.1016/0001-8708(69)90009-7 · Zbl 0192.49101 · doi:10.1016/0001-8708(69)90009-7
[15] Mosco, U. ”personal communication”.
[16] Skrypnik I.V., Math.Sbornik 184 pp 67– (1993)
[17] DOI: 10.5802/aif.204 · Zbl 0151.15401 · doi:10.5802/aif.204
[18] Ziemer, W.P. ”Weakly differentiable functions”. · Zbl 0692.46022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.