Periodic elliptic operators with asymptotically preassigned spectrum
Andrii Khrabustovskyi · Asymptotic Analysis · 2013
We deal with operators in R n of the form A=−1/b(x)Σ k=1 n ∂/∂x k (a(x)∂/∂x k ), where a(x),b(x) are positive, bounded and periodic functions. We denote by L per the set of such operators. The main result of this work is as follows: for an arbitrary L>0 and for arbitrary pairwise disjoint intervals (α j ,β j )⊂[0,L], j=1,…,m (m∈N) we construct the family of operators {A ε ∈L per } ε such that the spectrum of A ε has exactly m gaps in [0,L] when ε is small enough, and these gaps tend to the intervals (α j ,β j ) as ε→0. The idea how to construct the family {A ε } ε is based on methods of the homogenization theory.