Locally Periodic Unfolding Method and Two-Scale Convergence on Surfaces of Locally Periodic Microstructures

Abstract

Abstract. In this paper we generalize the periodic unfolding method and the notion of twoscale convergence on surfaces of periodic microstructures to locally periodic situations. The methods that we introduce allow us to consider a wide range of nonperiodic microstructures, especially to derive macroscopic equations for problems posed in domains with perforations distributed nonperiodically. Using the methods of locally periodic two-scale convergence on oscillating surfaces and the locally periodic boundary unfolding operator, we are able to analyze differential equations defined on boundaries of nonperiodic microstructures and consider nonhomogeneous Neumann conditions on the boundaries of perforations, distributed nonperiodically.

Multiscale Modelling and Analysis of Signalling Processes in Tissues with Non-Periodic Distribution of Cells

“…Also, using the same arguments as in [30] we obtain that c ε (t, x) ≥ 0 for (t, x) ∈ * ε,T and r ε l (t, x) ≥ 0 for (t, x) ∈ ε T , with l = f, b. To derive a priori estimates, we consider the structure of the microscopic equations.…”

“…In a similar way as in [9,21,30], we can prove the existence, uniqueness, and a priori estimates for a weak solution of problem (1)- (2). Notice that for the derivation of a priori estimates a trace estimate, uniform in ε, for functions…”

“…Proof (Sketch) As in [30] the existence of a solution of the microscopic problem (1) and (2) for each fixed ε > 0 is obtained by applying fixed point arguments and Galerkin method. Also, using the same arguments as in [30] we obtain that c ε (t, x) ≥ 0 for (t, x) ∈ * ε,T and r ε l (t, x) ≥ 0 for (t, x) ∈ ε T , with l = f, b.…”

“…Then, the convergence results for the l-p unfolding operator and l-t-s convergence, see [29,30] or Appendix, imply that there exist subsequences (denoted again by c ε , r ε f and r ε b ) and the functions c ∈ L 2 (0, T ;…”

“…We recall here the definition of locally periodic two-scale (l-t-s) convergence and l-p unfolding operator, see [29,30] for details. where L ε is the l-p approximation of ψ defined in (17).…”