Geometry Seminar: E. Schrohe (Leibniz U Hannover)
Geometry Seminar (Geometric Analysis)
Speaker: Elmar Schrohe (Leibniz University Hannover)
Title: The Plasmonic Eigenvalue Problem and the Dirichlet-to-Neumann Operator on Manifolds with Fibered Cusps
Speaker: Elmar Schrohe (Leibniz University Hannover)
Title: The Plasmonic Eigenvalue Problem and the Dirichlet-to-Neumann Operator on Manifolds with Fibered Cusps
Abstract: A plasmon of a bounded domain Ω ⊆ ℝⁿ is a nontrivial harmonic function on ℝⁿ \ ∂Ω that is continuous at ∂Ω and whose interior and exterior normal derivative at ∂Ω have a constant ratio. This ratio is called a plasmonic eigenvalue of Ω. It is indeed an eigenvalue of N₊⁻¹N₋, where N± denote the exterior and interior Dirichlet-to-Neumann operators.
Motivated by the case of two touching convex domains in ℝⁿ, we consider this problem on a manifold with fibered cusp singularities. In a first step we show that the Calderòn projector for elliptic operators in this setting is a matrix of φ-pseudodifferential operators in the sense of Mazzeo and Melrose. From this we derive that also the Dirichlet-to-Neumann operator is a first order φ-pseudodifferential operator. This gives us a precise understanding of the behavior of its Schwartz kernel near the boundary.
Joint work with Karsten Fritzsch and Daniel Grieser.