Homogenization of Elliptic Problems in Thick Junctions with Sharp Edges

Abstract

We consider the case when the thin discs of a thick multilevel junction can have sharp edges, i.e., their thickness tends to zero polynomially while approaching the edges. Three qualitatively different cases in the asymptotic behavior of the solution to a linear elliptic boundary-value problem are discovered depending on the edge form, namely rounded edges; linear wedges; and very sharp edges (in this case the boundary is not Lipshitz). Nonstandard and new techniques are proposed to get the corresponding homogenized problems (untypical in the last two cases). The obtained results mathematically justify an interesting physical effect for heat radiators (see conclusions to this chapter).