Quantitative analysis for $L^2$-estimates in linear elliptic equations via divergence-free transformation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918303184715776 |
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| author | Lee, Haesung |
| author_facet | Lee, Haesung |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04940 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative analysis for $L^2$-estimates in linear elliptic equations via divergence-free transformation Lee, Haesung Analysis of PDEs Numerical Analysis Primary: 35B45, 35J25, Secondary: 65N15, 68T07 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. |
| title | Quantitative analysis for $L^2$-estimates in linear elliptic equations via divergence-free transformation |
| topic | Analysis of PDEs Numerical Analysis Primary: 35B45, 35J25, Secondary: 65N15, 68T07 |
| url | https://arxiv.org/abs/2507.04940 |