{
  "id": "2408.12295",
  "version": "v1",
  "published": "2024-08-22T11:12:42.000Z",
  "updated": "2024-08-22T11:12:42.000Z",
  "title": "Analysis of linear elliptic equations with general drifts and $L^1$-zero-order terms",
  "authors": [
    "Haesung Lee"
  ],
  "comment": "The first version, 27 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-22T11:12:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J25",
      "35R05",
      "35B65",
      "35B35"
    ],
    "keywords": [
      "linear elliptic equations",
      "zero-order term",
      "general drifts",
      "dirichlet boundary value problem",
      "weak divergence-free drifts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}