Homogenization and concentrated capacity for the heat equation with non-linear variational data in reticular almost disconnected structures and applications to visual transduction

Out of a right, circular cylinder of height H and cross-section a disc of radius R+ one removes a stack of nH/ parallel, equi-spaced cylinders Cj,j=1,2,...,n, each of radius R and height . Here , are fixed positive numbers and is a positive parameter to be allowed to go to zero. The union of the Cj almost fills in the sense that any two contiguous cylinders Cj are at a mutual distance of the order of and that the outer shell, i.e., the gap S=-o has thickness of the order of (o is obtained from by formally setting =0). The cylinder from which the Cj are removed, is an almost disconnected structure, it is denoted by , and it arises in the mathematical theory of phototransduction. For each >0 we consider the heat equation in the almost disconnected structure , for the unknown function u, with variational boundary data on the faces of the removed cylinders Cj. The limit of this family of problems as 0 is computed by concentrating heat capacity and diffusivity on the outer shell, and by homogenizing the u within the limiting cylinder o. It is shown that the limiting problem consists of an interior diffusion in o and a boundary diffusion on the lateral boundary S of o. The interior diffusion is governed by the 2-dimensional heat equation in 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 uS. 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 uS. As a consequence of the geometry of , local arguments do not suffice to prove convergence in o, and also we have to take into account the behavior of the solution in S. A key, novel idea consists in extending equi-bounded and equi-Hölder continuous functions in -dependent domains, into equi-bounded and equi-Hölder continuous functions in the whole 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