On Stability of Square Root Domains for Non-Self-Adjoint Operators Under Additive Perturbations

TL;DR

This paper establishes the stability of square root domains for m-accretive operators under additive perturbations using a resolvent method, with applications to elliptic PDEs with complex coefficients and boundary conditions.

Contribution

It introduces a resolvent-based approach to prove stability of square root domains for non-self-adjoint operators under additive perturbations, extending to PDE applications.

Findings

Proves stability of square root domains under additive perturbations for m-accretive operators.

Establishes conditions for equality of domains involving the operator and its adjoint.

Applies the theoretical results to elliptic PDEs with nonsmooth coefficients and various boundary conditions.

Abstract

Assuming $T_0$ to be an m-accretive operator in the complex Hilbert space $\mathcal{H}$, we use a resolvent method due to Kato to appropriately define the additive perturbation $T = T_0 + W$ and prove stability of square root domains, that is, $$ dom\big((T_0 + W)^{1/2}\big) = dom\big(T_0^{1/2}\big). $$ Moreover, assuming in addition that $dom\big(T_0^{1/2}\big) = dom\big((T_0^*)^{1/2}\big)$, we prove stability of square root domains in the form $$dom\big((T_0 + W)^{1/2}\big) = dom\big(T_0^{1/2}\big) = dom\big((T_0^*)^{1/2}\big) = dom\big(((T_0 + W)^*)^{1/2}\big), $$ which is most suitable for PDE applications. We apply this approach to elliptic second-order partial differential operators of the form $$ - div(a\nabla \, \cdot \,) + \big(\mathbf{B}_1\cdot \nabla \cdot \big) + div \big(\mathbf{B}_2 \cdot \big) + V $$ in $L^2(\Omega)$ on certain open sets $\Omega \subseteq \mathbb{R}^n$, $n \in \mathbb{N}$, with Dirichlet, Neumann, and mixed boundary conditions on $\partial \Omega$, under general hypotheses on the (typically, nonsmooth, unbounded) coefficients and on $\partial\Omega$.