If the boundary is Lipschitz, one can prove only that the operator is bounded, it is not necessarily even compact. If the surface is smooth the operator is in fact a pseudodifferential operator of order -1, and Weyl asymptotics gives the answer. The latter result holds if the boundary is just a tiny little bit better than Lipschitz, but a hard analysis and perturbation theory is neded.
Grigori Rozenblum
- 31
- 2