Homogenization with Small Perforations of Increasingly Complicated Shapes
Alain Damlamian, Patrizia Donato · SIAM Journal on Mathematical Analysis · 1991
The limit case for a domain perturbed with many small and periodically distributed inclusions (holes or cracks) with increasingly complicated shapes is studied for the Poisson problem. On each “perforation” the boundary condition relates the value of the unknown function, assumed to be constant, with the total flux of the same unknown. The limit problem involves an extra term of order zero whose coefficient depends on all the parameters, in particular the capacity of the normalized inclusion and the measure of its boundary (or their behaviors as the size of the mesh goes to zero). Some examples involving fractal sets and cracks are given.