Analysis of linear elliptic equations with general drifts and $L^1$-zero-order terms
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915182631976960 |
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| author | Lee, Haesung |
| author_facet | Lee, Haesung |
| contents | This paper provides a detailed analysis of the Dirichlet boundary value problem for linear elliptic equations in divergence form with $L^p$-general drifts, where $p \in (d, \infty)$, and non-negative $L^1$-zero-order terms. Specifically, by transforming the general drifts into weak divergence-free drifts, we establish the existence and uniqueness of a bounded weak solution, showing that the zero-order term does not influence the quantity of the unique weak solution. Additionally, by imposing the $ VMO$ condition and mild differentiability on the diffusion coefficients and assuming an $L^s$-zero-order terms with $s \in (1, \infty)$, we demonstrate the existence and uniqueness of a strong solution for the corresponding non-divergence type equations. An important feature of this paper is that, due to the weak divergence-free property of the drifts in the transformed equations, the constants appearing in our estimates can be explicitly calculated, which is expected to offer significant applications in error analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12295 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analysis of linear elliptic equations with general drifts and $L^1$-zero-order terms Lee, Haesung Analysis of PDEs Primary: 35J25, 35R05, Secondary: 35B65, 35B35 This paper provides a detailed analysis of the Dirichlet boundary value problem for linear elliptic equations in divergence form with $L^p$-general drifts, where $p \in (d, \infty)$, and non-negative $L^1$-zero-order terms. Specifically, by transforming the general drifts into weak divergence-free drifts, we establish the existence and uniqueness of a bounded weak solution, showing that the zero-order term does not influence the quantity of the unique weak solution. Additionally, by imposing the $ VMO$ condition and mild differentiability on the diffusion coefficients and assuming an $L^s$-zero-order terms with $s \in (1, \infty)$, we demonstrate the existence and uniqueness of a strong solution for the corresponding non-divergence type equations. An important feature of this paper is that, due to the weak divergence-free property of the drifts in the transformed equations, the constants appearing in our estimates can be explicitly calculated, which is expected to offer significant applications in error analysis. |
| title | Analysis of linear elliptic equations with general drifts and $L^1$-zero-order terms |
| topic | Analysis of PDEs Primary: 35J25, 35R05, Secondary: 35B65, 35B35 |
| url | https://arxiv.org/abs/2408.12295 |