Diffusion on surfaces and the boundary periodic unfolding operator with an application to carcinogenesis in human cells (Q2927877)

The authors describe a homogenization problem which accounts for the evolution of a human cell within the carcinogenesis context. They consider a bounded domain \(\Omega \subset \mathbb{R}^{n}\) which represents the human cell and whose Lipschitz boundary \(\partial \Omega \) is supposed to be representable by a finite union of axis-parallel cuboids with corner coordinates in \(\mathbb{Q}^{n}\). The domain \(\Omega \) is supposed to be given a periodic structure, each \(\varepsilon \)-cell containing an inclusion outside which Benzo[a]pyrene molecules are present with a concentration \( u_{\varepsilon }\). The authors introduce the evolution equation \(\partial _{t}u_{\varepsilon }-D_{u}\Delta u_{\varepsilon }=-f(u_{\varepsilon })\) in the subdomain \(\Omega _{\varepsilon }\) of \(\Omega \). Here \(D_{u}\) is some positive diffusion coefficient, and \(f(x)\) is equal to 0 if \(x\leq 0\) and is equal to \(\frac{x}{x+M}Ma\) otherwise, for positive \(M,a\). They add boundary conditions for \(u_{\varepsilon }\) considering different parts of the overall boundary of \(\Omega _{\varepsilon }\). They also introduce a concentration \( v_{\varepsilon }\) of Benzo[a]pyrene-7,8-diol-9,10-epoxide molecules, which evolves in \(\Omega _{\varepsilon }\) according to the quite similar equation \( \partial _{t}v_{\varepsilon }-D_{v}\Delta v_{\varepsilon }=-g(v_{\varepsilon })\), where \(g\) has a similar structure than \(f\). They finally consider evolution equations for the concentration of these molecules on the boundary \(\Gamma _{\varepsilon }\), which involve the Laplace-Beltrami operator, and for the receptors. Once the whole system has been built, the authors introduce the appropriate functional spaces and weak formulations and they prove an existence and uniqueness result. The main result of the paper describes the asymptotic behavior of the solution of this evolution coupled system when the structure parameter goes to zero. The main tool of the proof is the unfolding method introduced by \textit{D. Cioranescu} et al. [C. R., Math., Acad. Sci. Paris 335, No. 1, 99--104 (2002; Zbl 1001.49016)] and in subsequent papers. The authors recall the main ideas of this useful tool and its main properties that they develop in the present context. They finally study some properties of the limit problem.