Non-periodic boundary homogenization and "light" concentrated masses
Gregory Aleksandrovich Chechkin, María‐Eugenia Pérez‐Martínez, Ekaterina Ivanovna Yablokova · Indiana University Mathematics Journal · 2005
We consider certain spectral problems for the Laplace operator with rapidly alternating boundary conditions in an open bounded domain n of R that contains many concentrated masses B e near the boundary. The regions B e have a diameter O(e) and the density takes the value e -m in BE and 1 outside. m, n and e are parameters: 0 ≤ m < 2, n ≥ 3 and e → 0. We assume small mass of the whole concentrated masses while periodicity of the microstructure is not assumed. We study the asymptotic behavior, as e → 0, of the eigenelements of the spectral problems. We obtain the homogenized (limit) spectral problems and estimates for the convergence rates of the corresponding eigenelements. Certain associated stationary problems are also considered, and estimates for the convergence rates of the solutions are obtained.