Quantitative analysis for $L^2$-estimates in linear elliptic equations via divergence-free transformation
Haesung Lee
TL;DR
This work provides an explicit $L^2$-estimate for weak solutions to linear elliptic equations in divergence form with general coefficients and an external source term. Using a divergence-free transformation, it converts the problem into an equation with a divergence-free drift, enabling a computable constant $C$ in the estimate $\|u\|_{L^2(U)} \le C \|f\|_{L^2(U)}$. The main result delivers an explicit $C$ that decreases as the diffusion coefficient or the zero-order term grows and remains robust even when the zero-order term is absent, with direct implications for a posteriori error bounds in Physics-Informed Neural Networks (PINNs). The analysis combines energy estimates, a tailored weight $\rho$, and a specialized Poincaré-type inequality, providing quantitative controls essential for practical PDE-aware learning and computational methods.
Abstract
This paper establishes an explicit $L^2$-estimate for weak solutions $u$ to linear elliptic equations in divergence form with general coefficients and external source term $f$, stating that the $L^2$-norm of $u$ over $U$ is bounded by a constant multiple of the $L^2$-norm of $f$ over $U$. In contrast to classical approaches based on compactness arguments, the proposed method, which employs a divergence-free transformation method, provides a computable and explicit constant $C>0$. The $L^2$-estimate remains robust even when there is no zero-order term, and the analysis further demonstrates that the constant $C>0$ decreases as the diffusion coefficient or the zero-order term increases. These quantitative results provide a rigorous foundation for applications such as a posteriori error estimates in Physics-Informed Neural Networks (PINNs), where explicit error bounds are essential.