Oblique and Normal Transmission Problems for Dirac Operators with Strongly Lipschitz Interfaces

Abstract

We investigate transmission problems with strongly Lipschitz interfaces for the Dirac equation by establishing spectral estimates on an associated boundary singular integral operator, the rotation operator. Using Rellich estimates we obtain angular spectral estimates on both the essential and full spectrum for general bi-oblique transmission problems. Specializing to the normal transmission problem, we investigate transmission problems for Maxwell's equations using a nilpotent exterior/interior derivativeoperator. The fundamental commutation properties for this operator with the two basic reflection operators are proved. We show how the L ₂spectral estimates are inherited for the domain of the exterior/interior derivative operator and prove some complementary eigenvalue estimates. Finally we use a general algebraic theorem to prove a regularity property needed for Maxwell's equations.

Keywords:

• Math Subject Classification:35J55
• 35Q60
• 45E05

Acknowledgments

The results in this article have been obtained during my Ph.D. studies at the Centre for Mathematics and Its Applications at the Australian National University. I want to thank my supervisor Alan McIntosh for generously sharing his mathematical insights with me.