Remarks on well-posedness for linear elliptic equations via divergence-free transformation
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918308081565696 |
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| author | Lee, Haesung |
| author_facet | Lee, Haesung |
| contents | This paper investigates the well-posedness of linear elliptic equations, focusing on the divergence-free transformation introduced in the author's recent work [J. Math. Anal. Appl. 548 (2025), 129425]. By comparing this approach with classical bilinear form methods, we demonstrate that while standard techniques encounter limitations in handling zero-order coefficients $c \in L^1(U)$, the divergence-free transformation successfully establishes well-posedness in this setting. Furthermore, utilizing the Riesz-Thorin interpolation theorem between the cases $c \in L^1(U)$ and $c \in L^{\frac{2d}{d+2}}(U)$, we establish the existence and uniqueness of weak solutions under the assumption $c \in L^s(U)$ for $s \in [1, \frac{2d}{d+2}]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19317 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Remarks on well-posedness for linear elliptic equations via divergence-free transformation Lee, Haesung Analysis of PDEs Functional Analysis Primary 35J15, 35J25, Secondary 35J75, 35B45, 46B70 This paper investigates the well-posedness of linear elliptic equations, focusing on the divergence-free transformation introduced in the author's recent work [J. Math. Anal. Appl. 548 (2025), 129425]. By comparing this approach with classical bilinear form methods, we demonstrate that while standard techniques encounter limitations in handling zero-order coefficients $c \in L^1(U)$, the divergence-free transformation successfully establishes well-posedness in this setting. Furthermore, utilizing the Riesz-Thorin interpolation theorem between the cases $c \in L^1(U)$ and $c \in L^{\frac{2d}{d+2}}(U)$, we establish the existence and uniqueness of weak solutions under the assumption $c \in L^s(U)$ for $s \in [1, \frac{2d}{d+2}]$. |
| title | Remarks on well-posedness for linear elliptic equations via divergence-free transformation |
| topic | Analysis of PDEs Functional Analysis Primary 35J15, 35J25, Secondary 35J75, 35B45, 46B70 |
| url | https://arxiv.org/abs/2601.19317 |