Periodic elliptic operators with asymptotically preassigned spectrum

Abstract

We deal with operators in $\mathbb{R}^n$ of the form $$\mathbf{A}=-{1\over \mathbf{b}(x)}\sum\limits_{k=1}^n\ds{\partial\over\partial x_k}(\mathbf{a}(x){\partial \over\partial x_k})$$ where $\mathbf{a}(x),\mathbf{b}(x)$ are positive, bounded and periodic functions. We denote by $\mathbf{L}_{\mathrm{per}}$ 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 $(\alpha_j,\beta_j)\subset[0,L]$, $j=1,...,m$ ($m\in\mathbb{N}$) we construct the family of operators $\{\mathbf{A}^\varepsilon\in \mathbf{L}_{\mathrm{per}}\}_{\varepsilon}$ such that the spectrum of $\mathbf{A}^\varepsilon$ has exactly $m$ gaps in $[0,L]$ when $\varepsilon$ is small enough, and these gaps tend to the intervals $(\alpha_j,\beta_j)$ as $\varepsilon\to 0$. The idea how to construct the family ${\mathbf{A}\e}_\eps$ is based on methods of the homogenization theory.