On the Stability of Linear Elliptic Equations with $$L^2$$-Drifts of Negative Divergence and Singular Zero-Order Terms

This paper first demonstrates the existence and uniqueness of solutions to homogeneous Dirichlet boundary value problems for second-order linear elliptic equations with $$L^2$$ -drifts of negative divergence and positive $$L^1$$ -zero-order terms, based on a functional analytic approach, including weak convergence methods and duality arguments. By improving the previous contraction properties, which may not be effective when the zero-order term is very small, this paper introduces a general $$L^2$$ -“contraction” property for any positive zero-order term, leading to remarkable results regarding $$L^2$$ -stability. These stability results are applicable to $$L^2$$ -error analysis for physics-informed neural networks, and can also be applied to stationary Schrödinger operators with $$L^2$$ -zero-order terms. We emphasize that all the constants arising in the estimates of this paper can be explicitly computed.