Stability of the $L^p$ Poisson-Neumann problem under small perturbations of the coefficients and the domain

In this article, we obtain three stability results for the Poisson-Neumann problem: (1) the stability of the $L^p$ Poisson-Neumann problem under small Carleson perturbations of the coefficients, (2) the preservation of the $L^p$ Poisson-Neumann problem when transitioning from the Laplacian to operators in a special form with oscillations satisfying a small Carleson condition (''small special DKP operators''), and (3) the stability of the $L^p$ Poisson-Neumann problem under small bi-Lipschitz transformations of domains. In contrast to the Neumann problem, the Poisson-Neumann problem is notably more self-contained, as results pertaining to it do not require assumptions regarding the $L^{p'}$ Dirichlet problem, making it a more suitable object for a perturbation theory. As intermediate steps, we establish estimates on the square functional for solutions to the Poisson problem and their gradients on chord arc domains, which are of independent interest for specialists.