{
  "id": "1601.01626",
  "version": "v1",
  "published": "2016-01-07T18:04:39.000Z",
  "updated": "2016-01-07T18:04:39.000Z",
  "title": "$\\mathbb{B}$-valued monogenic functions and their applications to boundary value problems in displacements of 2-D Elasticity",
  "authors": [
    "S. V. Gryshchuk"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Consider the commutative algebra $\\mathbb{B}$ over the field of complex numbers with the bases $\\{e_1,e_2\\}$ such that %satisfying the conditions $(e_1^2+e_2^2)^2=0$, $e_1^2+e_2^2\\ne 0$. %$\\mathbb{B}$ is unique. Let $D$ be a domain in $xOy$, $D_{\\zeta}:=\\{xe_1+ye_2:(x,y) \\in D\\}\\subset \\mathbb{B}$. We say that $\\mathbb{B}$-valued function $\\Phi \\colon D_{\\zeta} \\longrightarrow \\mathbb{B}$, $\\Phi(\\zeta)=U_{1}\\,e_1+U_{2}\\,ie_1+ U_{3}\\,e_2+U_{4}\\,ie_2$, $\\zeta=xe_1+ye_2$, $U_{k}=U_{k}(x,y)\\colon D\\longrightarrow \\mathbb{R}$, $k=\\bar{1,4}$, is {\\em monogenic} in $D_{\\zeta}$ iff $\\Phi$ has the classic derivative in every point in $D_{\\zeta}$. Every $U_k$, $k=\\bar{1,4}$, is a biharmonic function in $D$. A problem on finding an elastic equilibrium for isotropic body $D$ by given boundary values on $\\partial D$ of partial derivatives $\\frac{\\partial u}{\\partial v}$, $\\frac{\\partial v}{\\partial y}$ for displacements $u$, $v$ is equivalent to BVP for monogenic functions, which is to find $\\Phi$ by given boundary values of $U_1$ and $U_4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-07T18:04:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30G35",
      "31A30",
      "74B05"
    ],
    "keywords": [
      "boundary value problems",
      "valued monogenic functions",
      "displacements",
      "elasticity",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160101626G"
    }
  }
}