{
  "id": "1904.04722",
  "version": "v1",
  "published": "2019-04-09T15:03:35.000Z",
  "updated": "2019-04-09T15:03:35.000Z",
  "title": "Regularity theory and Green's function for elliptic equations with lower order terms in unbounded domains",
  "authors": [
    "Mihalis Mourgoglou"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider elliptic operators in divergence form with lower order terms of the form $Lu = -\\text{div} (A \\cdot \\nabla u + b u ) - c\\cdot \\nabla u - du$, in an open set $\\Omega \\subset \\mathbb{R}^n$, $n \\geq 3$, with possibly infinite Lebesgue measure. We assume that the $n \\times n$ matrix $A$ is uniformly elliptic with real, merely bounded and possibly non-symmetric coefficients, $b, c \\in L^n(\\Omega)$, $d \\in L^\\frac{n}{2}(\\Omega)$, and $ \\mathcal{K}_{Dini,1/2}(\\Omega)$ stands for a variant of the Stummel-Kato class with a Dini-type condition. If $\\text{div} b +d \\leq 0$ holds, or if $-\\text{div} c + d \\leq 0$ and $|b+c|^2 \\in \\mathcal{K}_{Dini,1/2}(\\Omega)$ hold, then we develop a De Giorgi/Nash/Moser theory for solutions of $Lu = f - \\text{div} g$, where $ f$ and $|g|^2 \\in \\mathcal{K}_{Dini,1/2}(\\Omega)$. The most important feature of our proofs is that all the estimates are scale invariant and independent of $\\Omega$. We also show that the variational Dirichlet problem is well-posed and, for its solution, we prove a Wiener-type criterion for boundary regularity. Finally, assuming that $|b+c|^2 \\in \\mathcal{K}_{Dini,1/2}(\\Omega)$, we construct Green's functions associated with operators of the form above and show, among others, global pointwise bounds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-09T15:03:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A08",
      "35B50",
      "35B51",
      "35B65",
      "35J08",
      "35J15",
      "35J20",
      "35J25",
      "35J67",
      "35J86"
    ],
    "keywords": [
      "lower order terms",
      "regularity theory",
      "elliptic equations",
      "unbounded domains",
      "construct greens functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}