WIAS Preprint No. 1439, (2009)

Direct computation of elliptic singularities across anisotropic, multi-material edges


  • Haller-Dintelmann, Robert
  • Kaiser, Hans-Christoph
  • Rehberg, Joachim

2010 Mathematics Subject Classification

  • 35B65 35J25 35R05


  • Elliptic transmission problems, mixed boundary problems, $W^1,p$ regularity




We characterise the singularities of elliptic div-grad operators at points or edges where several materials meet on a Dirichlet or Neumann part of the boundary of a two- or three-dimensional domain. Special emphasis is put on anisotropic coefficient matrices. The singularities can be computed as roots of a characteristic transcendental equation. We establish uniform bounds for the singular values for several classes of three- and four-material edges. These bounds can be used to prove optimal regularity results for elliptic div-grad operators on three-dimensional, heterogeneous, polyhedral domains with mixed boundary conditions. We demonstrate this for the benchmark L--shape problem.

Appeared in

  • J. Math. Sci. (N. Y.), 172 (2011) pp. 589--622.

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