Stability of square root domains associated with elliptic systems of PDEs on nonsmooth domains

Abstract

We discuss stability of square root domains for uniformly elliptic partial differential operators $L_{a,\Omega ,\Gamma }=-\nabla \cdot a\nabla$ in $L^2(\Omega )$, with mixed boundary conditions on $\partial \Omega$, with respect to additive perturbations. We consider open, bounded, and connected sets $\Omega \in {\mathbb{R}}^n$, $n\in \mathbb{N}\backslash \{1\}$, that satisfy the interior corkscrew condition and prove stability of square root domains of the operator $L_{a,\Omega ,\Gamma }$ with respect to additive potential perturbations $V\in L^p(\Omega )+L^{\infty }(\Omega )$, $p>n/2$. Special emphasis is put on the case of uniformly elliptic systems with mixed boundary conditions.

13 pages, to appear in Journal of Differential Equations