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