Homogenization of the parabolic equation with periodic coefficients at the edge of a spectral gap

Abstract

In , consider a second-order elliptic differential operator , , of the form with periodic coefficients. For small ε, we study the behavior of the semigroup , t>0, cut by the spectral projection of the operator for the interval . Here is the right edge of a spectral gap for the operator . We obtain approximation for the ‘cut semigroup’ in the operator norm in with error , and also a more accurate approximation with error (after singling out the factor ). The results are applied to homogenization of the Cauchy problem , , with the initial data from a special class.