1305.1009 (Denis Borisov et al.)

Homogenization and uniform resolvent convergence for elliptic operators
in a strip perforated along a curve    [PDF]

Denis Borisov, Giuseppe Cardone, Tiziana Durante

We consider an infinite planar straight strip perforated along an infinite curve by small holes located closely one to another. In such domain we consider a general second order elliptic operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic and satisfies rather weak assumptions, we describe possible homogenized problems. Our main result is the proof of the uniform resolvent convergence of the perturbed operator to a homogenized one in various operator norms and the estimates for the rate of convergence. On the basis of the uniform convergence we show the convergence of the spectrum. In a particular case of pure periodic perforation and Dirichlet condition on the boundary of the holes we obtain two-terms asymptotics for the first band functions of the perturbed operator.

View original: http://arxiv.org/abs/1305.1009