Slides from the version given at the University of Linz, July 2006
The biharmonic double layer potential integral operator is a 2x2 system of strongly singular and hypersingular boundary integral operators. The corresponding integral equation of the second kind that can be used to solve the biharmonic Dirichlet problem (clamped plate problem). It is, however, not a Fredholm integral equation of the second kind, even on a smooth domain where the calculus of pseudodifferential operators can be used to show that it is elliptic. On non-smooth domains, there is not much available for analyzig this system. We show that Poincaré's energy estimates can be used to prove that the Neumann series for this integral equation converges in the energy norm, in close analogy to what is known for the Laplacian and, more generally, for second order strongly elliptic systems.
PDF (600 k)