# zbMATH — the first resource for mathematics

Asymptotic behaviour of a sequence of Neumann problems. (English) Zbl 0895.35011
The author studies the limit behaviour of Neumann problems on fractured and perforated domains (the excepted closed subsets of the domain “accumulate” on a surface). The main result states that the limit of solutions minimizes a functional whose Euler equation is the same Neumann problem but involving transmission conditions on the “accumulation” surface. The proof is based on $$\Gamma$$-convergence arguments and integral representation theorems.
Reviewer: C.Popa (Iaşi)

##### MSC:
 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35J25 Boundary value problems for second-order elliptic equations 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
$$\Gamma$$-convergence; transmission conditions
Full Text:
##### References:
 [1] Attouch H., Rend. Sem. Mat. Univ. Politec. Torino 45 pp 71– (1987) [2] Bucur D., Series on Advances in Mathematics for Applied Sciences 18, in: Homogeneization and Continuum Mechanics (CIRM – Luminy, Marseille, 1993) pp 117– (1994) [3] Bucur D., C.R. Acad. Sci. Paris, Série I 318 pp 795– (1994) [4] Bucur D., C.R. Acad. Sci. Paris, Série I 319 pp 57– (1994) [5] Choquet G., Mathematics Leature Notes Series (1969) [6] Cioranescu D., Research Notes in Mathematics 60, in: Collége de France Seminar, pp 98,154– (1982) [7] Dal Maso, G. 1993. ”An Introduction to {$$\Gamma$$}–Convergence, Birkh$$aacute;user". Boston$$ · Zbl 0816.49001 [8] Dal Maso G., J. Math. Pures Appl. 73 pp 1– (1994) [9] Damlamian A., Rend. Sem. Mat. Univ. Politec. Torino 43 pp 427– (1985) [10] De Giorgi E., Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 82 pp 842– (1975) [11] DOI: 10.1007/BF01762424 · Zbl 0635.35021 [12] DOI: 10.1512/iumj.1972.22.22013 · Zbl 0238.28015 [13] DOI: 10.1070/SM1970v012n04ABEH000938 · Zbl 0219.31010 [14] DOI: 10.1070/SM1979v035n02ABEH001474 · Zbl 0421.35019 [15] Marchenko, A.V. and Krushlov: E.Ya. 1974. ”Boundary value problems in domains with finely–granulated boundariesOn the Neumann boundary problem in a domain with complicated boundary,(in Russian)”. Naukova Dumka, Kiev [16] Murat: F., Research Notes in Mathematics 127 pp 24– (1985) [17] Nguetseng G., RAIRO Modél. Math. Anal. Nurnér 19 pp 33– (1985) [18] Picard C., RAIRO Modél. Math. Anal. Numér 21 pp 293– (1987) [19] Sanchez–palemcia E., Research Notes in Mathematics 70 pp 309– (1982) [20] Sanchez–Palencia E., Research Notes in Mathematics 125 pp 309– (1985) · Zbl 0604.76083 [21] Marchenko, A.V. 1989. ”Weakly Differentiable Functions,”. Edited by: Ziemer: W.P. Berlin: Springer–Verlag.
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.