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• W2023135664 abstract "Out of a right, circular cylinder Ωε of height H and cross-section a disc of radius R+σε one removes a stack of n≈H/ε parallel, equi-spaced cylinders C j , j=1,2,...,n, each of radius R and height νε. Here σ, ν are fixed positive numbers and ε is a positive parameter to be allowed to go to zero. The union of the C j almost fills Ωε in the sense that any two contiguous cylinders C j are at a mutual distance of the order of ε and that the outer shell, i.e., the gap S ε=Ωε-Ω o has thickness of the order of ε (Ω o is obtained from Ωε by formally setting ε=0). The cylinder Ωε from which the C j are removed, is an almost disconnected structure, it is denoted by Ωε, and it arises in the mathematical theory of phototransduction. For each ε>0 we consider the heat equation in the almost disconnected structure Ωε, for the unknown function u ε, with variational boundary data on the faces of the removed cylinders C j . The limit of this family of problems as ε→0 is computed by concentrating heat capacity and diffusivity on the outer shell, and by homogenizing the u ε within the limiting cylinder Ω o . It is shown that the limiting problem consists of an interior diffusion in Ω o and a boundary diffusion on the lateral boundary S of Ω o . The interior diffusion is governed by the 2-dimensional heat equation in Ω o , for an interior limiting function u. The boundary diffusion is governed by the Laplace–Beltrami heat equation on S, for a boundary limiting function u S . Moreover the exterior flux of the interior limit u provides the source term for the boundary diffusion on S. Finally the interior limit u, computed on S in the sense of the traces, coincides with the boundary limit u S . As a consequence of the geometry of Ωε, local arguments do not suffice to prove convergence in Ω o , and also we have to take into account the behavior of the solution in S ε. A key, novel idea consists in extending equi-bounded and equi-Hölder continuous functions in ε-dependent domains, into equi-bounded and equi-Hölder continuous functions in the whole ℝ N , by means of the Kirzbraun–Pucci extension technique. The biological origin of this problem is traced, and its application to signal transduction in the retina rod cells of vertebrates is discussed." @default.
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• W2023135664 date "2003-08-15" @default.
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• W2023135664 title "Homogenization and concentrated capacity for the heat equation with non-linear variational data in reticular almost disconnected structures and applications to visual transduction" @default.
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• W2023135664 doi "https://doi.org/10.1007/s10231-003-0072-6" @default.
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