Convergence towards the homogenized solution in case of lipschitzian boundary

Abstract

The aim of this paper is to prove, for solutions of periodic elliptic boundary value problems with nonsmooth boundary and discontinuous coefficients, the strong convergence to the corresponding homogenized solutions. Weak convergence estimates for such homogenization problems are well known and are obtained directly from G-convergence, like for instance in [V.V. Jikov et al. (1994). Homogenization of Differential Operators and Integral Functions. Springer Verlag, Berlin]. Convergence estimates in case of smooth coefficients are also classical, see for instance, in [A. Bensoussan, J.L. Lions and G. Papanicolaou (1978). Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam], [N.S. Bakhvalov and G. Panasenko (1989). Homogenization, Averaging processes in Periodic Media. Kluwer, Dordrecht] or [O.A. Oleinik, A.S. Shamaev and G.A. Yosifian (1992). Mathematical problems in elasticity and homogeni-zation. Studies in Mathematics and its Application, Elsevier, Amsterdam, 26]. The present article is extending the results of [N.S. Bakhvalov and A. Bourgeat (1998). Precise estimates of the difference between the homogenized solution, with its first corrector and the original one. Applicable Analysis, 70(1-2), 45-60], where more smoothness of the boundaries was assumed, namely boundaries belonging to C^3+λ.

Keywords:

• Elliptic PDEs;
• Homogenization;
• Convergence;
• Estimates;
• Regularity

AMS Classification:

• 35B30
• 35B10
• 35B27

^*Corresponding author. [email protected]

^† [email protected]

^*Corresponding author. [email protected]

^† [email protected]

Notes

^*Corresponding author. [email protected]

^† [email protected]