{
  "id": "2405.09540",
  "version": "v1",
  "published": "2024-05-15T17:53:32.000Z",
  "updated": "2024-05-15T17:53:32.000Z",
  "title": "Singular parabolic operators in the half-space with boundary degeneracy: Dirichlet and oblique derivative boundary conditions",
  "authors": [
    "Luigi Negro"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study elliptic and parabolic problems governed by the singular elliptic operators $$ \\mathcal L=y^{\\alpha_1}\\mbox{Tr }\\left(QD^2_x\\right)+2y^{\\frac{\\alpha_1+\\alpha_2}{2}}q\\cdot \\nabla_xD_y+\\gamma y^{\\alpha_2} D_{yy}+y^{\\frac{\\alpha_1+\\alpha_2}{2}-1}\\left(d,\\nabla_x\\right)+cy^{\\alpha_2-1}D_y-by^{\\alpha_2-2}$$ in the half-space $\\mathcal{R}^{N+1}_+=\\{(x,y): x \\in \\mathcal{R}^N, y>0\\}$, under Dirichlet or oblique derivative boundary conditions. In the special case $\\alpha_1=\\alpha_2=\\alpha$ the operator $\\mathcal L$ takes the form $$ \\mathcal L=y^{\\alpha}\\mbox{Tr }\\left(AD^2\\right)+y^{\\alpha-1}\\left(v,\\nabla\\right)-by^{\\alpha-2},$$ where $v=(d,c)\\in\\mathcal{R}^{N+1}$, $b\\in\\mathcal{R}$ and $ A=\\left( \\begin{array}{c|c} Q & { q}^t \\\\[1ex] \\hline q& \\gamma \\end{array}\\right)$ is an elliptic matrix. We prove elliptic and parabolic $L^p$-estimates and solvability for the associated problems. In the language of semigroup theory, we prove that $\\mathcal L$ generates an analytic semigroup, characterize its domain as a weighted Sobolev space and show that it has maximal regularity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-15T17:53:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K67",
      "35B45",
      "47D07",
      "35J70",
      "35J75"
    ],
    "keywords": [
      "oblique derivative boundary conditions",
      "singular parabolic operators",
      "boundary degeneracy",
      "half-space",
      "singular elliptic operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}