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

TL;DR

This paper investigates the stability of square root domains for elliptic PDE operators with mixed boundary conditions under additive perturbations, emphasizing the case of elliptic systems on nonsmooth domains.

Contribution

It establishes the stability of square root domains for elliptic operators with mixed boundary conditions under potential perturbations on nonsmooth domains.

Findings

Stability of square root domains proven for elliptic operators with mixed boundary conditions.

Results apply to nonsmooth, corkscrew domains in higher dimensions.

Perturbation stability holds for potentials in L^p + L^∞ with p > n/2.

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.