Homogenization of parabolic equation in an evolving domain with an oscillating boundary

Homogenization of the heat equation in a noncylindrical domain with randomly oscillating boundary

“…For the homogenization, it is enough to consider homogeneous boundary condition on Γ 1 × (0, T) due to the lifting of parabolic boundary data (cf. Muthukumar et al 1 ). We rewrite (4.2) in the following form with homogeneous boundary condition:…”

“…The problem to be well-posed, we have to get uniform ellipticity of A 𝜀 . This issue has been addressed in the recent article 1 for the deterministic case with a slightly different matrix, namely deterministic matrix 1…”

“…21 They used the boundary layer techniques to derive the effective problem and the order of convergence. The Navier-Stokes system on a cylindrical domain with an oscillating boundary of amplitude O(𝜀) was studied in Bucur et al 16 There are many results on the boundary value problems on oscillating domains with a fixed amplitude, that is, the amplitude of O (1). Brizzi and Chalot 22 studied the homogenization of the Laplace equation with homogeneous Neumann boundary conditions in domains with the periodic oscillating boundary of fixed amplitude.…”

“…15,24 The asymptotic analysis of optimal control problems was considered in other works. [25][26][27] In Aiyappan et al, 12 the generalized unfolding method is used to derive the homogenization problem on oscillating circular boundary domain with amplitude of O (1).…”

“…Motivated by the Muthukumar et al's work, 1 the aim of this article is to present the homogenization procedure of the problem (1.1). However, the combination of the oscillating boundary and its complex geometry, our problem leads to additional mathematical difficulties to deal with.…”