Quantitative analysis for $L^2$-estimates in linear elliptic equations via divergence-free transformation

TL;DR

This paper derives explicit $L^2$-estimates for solutions to linear elliptic equations using divergence-free transformations, enabling computable bounds crucial for applications like PINNs.

Contribution

It introduces a divergence-free transformation method to obtain explicit, computable $L^2$-estimates for elliptic equations, improving upon classical compactness-based approaches.

Findings

The $L^2$-norm of solutions is bounded by a constant times the $L^2$-norm of the source term.

The constant decreases as diffusion or zero-order terms increase.

The method applies even without zero-order terms, providing robust estimates.

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.