(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Haller-Dintelmann, Robert; Kaiser, Hans-Christoph; Rehberg, Joachim
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 fourmaterial
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
Lshape problem.
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Haller-Dintelmann, Robert; Rehberg, Joachim
We show that usual second order operators in divergence form satisfy
coercivity on Lipschitz domains if they are either complemented with
homogeneous Dirichlet boundary conditions on a set of non-zero boundary
measure or if a suitable Robin boundary condition is posed. Moreover, we
prove the positivity of solutions in a general, abstract setting, provided
that the right hand side is a positive functional. Finally, positive elements
from $W^-1,2$ are identified as positive measures
