Homogenization results for a class of parabolic equations with a non-local interface condition via time-periodic unfolding

Abstract

We study the thermal properties of a composite material in which a periodic array of finely mixed perfect thermal conductors is inserted. The suitable model describing the behaviour of such physical materials leads to the so-called equivalued surface boundary value problem. To analyze the overall conductivity of the composite medium (when the size of the inclusions tends to zero), we make use of the homogenization theory, employing the unfolding technique. The peculiarity of the problem under investigation asks for a particular care in developing the unfolding procedure, giving rise to a non-standard two-scale problem.

Homogenization of a Heat Conduction Problem with a Total Flux Boundary Condition

“…The proof can be obtained as in the first part of the proof of Theorem 4.12 in [5]. We stress that, due to the positive definiteness of the matrix A hom , equation 4.23has a unique solution.…”

“…In order to provide the single-scale problem satisfied by the homogenized function u, we state the following Lemma, whose proof can be found in [5].…”

“…Nevertheless, by a standard factorization procedure, we can decouple this two-scale problem, thus obtaining a parabolic equation with a uniformly elliptic principle part, ensuring the uniqueness of the solution. A generalized version of this problem, involving time-oscillating coefficients, is treated by a completely different approach in [5].…”

Diffusion in inhomogeneous media with periodic microstructures

“…In this Section, we always assume that 𝛼 ≤ 1 and 𝑣 𝜀 is the solution to (21), under the assumptions listed in Subsection 2.2.…”

“…At our knowledge, diffusion problems governed by Fick and/or Fokker-Planck laws depending on capacitive, diffusive and Fokker coefficients, highly oscillating with respect to time and space simultaneously are not considered in an extensive body of mathematical literature. Among the few results, we recall [11,19,[21][22][23][24][25][26].…”