On Strong Closure of Sets of Feasible States Associated with Families of Elliptic Operators

Abstract

The closure of sets of feasible states for systems of elliptic equations in the strong topology of the Cartesian product $[H^1_0 (\Omega)]^m$ of Sobolev spaces is considered. For $m = 2$ and $\Omega \subset \mathbb R^2$, it is shown that there is a family of linear elliptic operators of the type div $(\chi \mathcal A^1 + (1 - \chi)\mathcal A^2)\triangledown$, where $\chi$ belongs to the set of all characteristic functions of measurable subsets of $\Omega$, such that there does not exist a larger family of operators of the type div $\mathcal A \triangledown$ for which the sets of feasible states coincide with the closure of the original ones.