The one‐dimensional model for an elliptic equation in a perforated thin anisotropic heterogeneous three‐dimensional structure.

In this paper, we investigate the one‐dimensional model for a thin three‐dimensional structure Ω^ε$$ {\hat{\Omega}}_{\varepsilon } $$ in the framework of the thermal conduction. The structure is characterized by two small positive parameters ε$$ \varepsilon $$ and rε$$ {r}_{\varepsilon } $$. The first parameter ε$$ \varepsilon $$ corresponds to the thickness of the structure while the second one characterizes the thickness of its core Tε$$ {T}_{\varepsilon } $$ which plays the role of a "hole." The structure is assumed to be heterogeneous and anisotropic, and we deal with three cases related to the limit limε→0ε2|ln(rε)|=k,k∈{0,1,+∞}$$ \underset{\varepsilon \to 0}{\lim}\kern.3em {\varepsilon}^2\mid \ln \left({r}_{\varepsilon}\right)\mid =k,k\in \left\{0,1,+\infty \right\} $$. We exhibit the "strange" term appearing in the one‐dimensional model in the critical case k=1$$ k=1 $$, and we highlight the effect of the anisotropy on the form of the corrector for uε$$ {u}_{\varepsilon } $$. [ABSTRACT FROM AUTHOR]