{
  "abstract": "In this paper, we study the existence, uniqueness, and quantitative estimates for weak solutions to linear elliptic Dirichlet problems of the form \\[ -\\operatorname{div}(γ\\nabla u)+\\langle \\nablaϕ+\\mathbf{H},\\nabla u\\rangle+(c+α)u=f \\quad\\text{ in }U, \\quad\\; u=0 \\quad\\text{on }\\partial U, \\] where $U\\subset \\mathbb{R}^d$ is bounded, $γ\\in[1,\\infty)$ is a constant, $ϕ\\in H^{1,2}(U)\\cap L^\\infty(U)$, $\\mathbf{H}\\in L^p(U,\\mathbb{R}^d)$ for some $p \\in (d, \\infty)$, and $c\\in L^1(U)$ with $c\\ge0$. A key feature of this setting is that the drift contains the low-regularity term $\\nablaϕ$, which is only assumed to belong to $L^2(U,\\mathbb{R}^d)$, while the zero-order coefficient is merely integrable. We prove that, even under these rough assumptions, well-posedness and quantitative energy and $L^2$ estimates remain valid. In addition, by using a previously established interpolation result, we characterize a trade-off between the integrability of the source term $f$ and that of the zero-order coefficient $c$, and show that well-posedness together with the corresponding quantitative estimates hold under these interpolated assumptions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06227",
  "author_total": 1,
  "authors": [
    {
      "name": "Haesung Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2xfz87mq66000.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06227",
  "fulltext_url": null,
  "id": "2xfz87mq66000",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06227",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7212060547",
  "pdf_url": "https://arxiv.org/pdf/2609.06227",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2xfz87mq66000.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06227",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06227"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Notes on linear elliptic equations with $L^2$-gradient perturbations and singular zero-order coefficients",
  "title_text": "Notes on linear elliptic equations with L^2-gradient perturbations and singular zero-order coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/2xfz87mq66000",
  "venue": "arXiv",
  "versions": []
}