×

zbMATH — the first resource for mathematics

Random homogenization of an obstacle problem. (English) Zbl 1180.35069
This paper is devoted to the homogenization process of the obstacle problem associated to a perforated domain in the case when the holes are contained in a set of periodically distributed balls of critical size in the sense of [D. Cioranescu and F. Murat, Prog. Nonlinear Differ. Equ. Appl. 31, 45–93 (1997; Zbl 0912.35020)]. While the shape of the holes remains unspecified, the rescaled (with respect to the size of the periodic cell) capacity of every hole is assumed to be given by a stationary ergodic process. In order to prove the convergence result, the authors introduce an auxiliary obstacle problem and appropriately construct a fine corrector.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
49J40 Variational inequalities
47A35 Ergodic theory of linear operators
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Akcoglu, M.A.; Krengel, U., Ergodic theorems for superadditive processes, J. reine angew. math., 323, 53-67, (1981) · Zbl 0453.60039
[2] Carbone, L.; Colombini, F., On convergence of functionals with unilateral constraints, J. math. pures appl. (9), 59, 4, 465-500, (1980) · Zbl 0415.49010
[3] Caffarelli, L.A.; Souganidis, P.E.; Wang, L., Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media, Comm. pure appl. math., 58, 3, 319-361, (2005) · Zbl 1063.35025
[4] Cioranescu, D.; Murat, F., Un terme étrange venu d’ailleurs, (), 98-138, 389-390 · Zbl 0496.35030
[5] Cioranescu, D.; Murat, F., Un terme étrange venu d’ailleurs. II, (), 154-178, 425-426 · Zbl 0496.35030
[6] Dal Maso, G., Asymptotic behaviour of minimum problems with bilateral obstacles, Ann. mat. pura appl. (4), 129, 327-366, (1981)
[7] Dal Maso, G.; Longo, P., γ-limits of obstacles, Ann. mat. pura appl. (4), 128, 1-50, (1981) · Zbl 0467.49004
[8] Dal Maso, G.; Modica, L., Nonlinear stochastic homogenization and ergodic theory, J. reine angew. math., 368, 28-42, (1986) · Zbl 0582.60034
[9] De Giorgi, E.; Dal Maso, G.; Longo, P., γ-limits of obstacles, Atti accad. naz. lincei rend. cl. sci. fis. mat. natur. (8), 68, 6, 481-487, (1980) · Zbl 0512.49011
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.