2025/07/07 by Lee, Haesung
#35J25 #68T07 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Primary: 35B45 #Secondary: 65N15
paper · doi:10.48550/arxiv.2507.04940
This paper establishes an explicit L2-estimate for weak solutions u to linear elliptic equations in divergence form with general coefficients and external source term f, stating that the L2-norm of u over U is bounded by a constant multiple of the L2-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 L2-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.