On vanishing near corners of transmission eigenfunctions

By E. Blaasten, Hongyu Liu. 2017

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Abstract Let Ω be a bounded domain in R n , n ≥ 2 , and V ∈ L ∞ ( Ω ) be a potential function. Consider the following transmission eigenvalue problem for nontrivial v , w ∈ L 2 ( Ω ) and k ∈ R + , { ( Δ + k 2 ) v = 0 in Ω , ( Δ + k 2 ( 1 + V ) ) w = 0 in Ω , w − v ∈ H 0 2 ( Ω ) , ‖ v ‖ L 2 ( Ω ) = 1 . We show that the transmission eigenfunctions v and w carry the geometric information of supp ( V ) . Indeed, it is proved that v and w vanish near a corner point on ∂Ω in a generic situation where the corner possesses an interior angle less than π and the potential function V does not vanish at the corner point. This is the first quantitative result concerning the intrinsic property of transmission eigenfunctions and enriches the classical spectral theory for Dirichlet/Neumann Laplacian. We also discuss its implications to inverse scattering theory and invisibility.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7c7f6d76894cd1fe2489219253c0c869936a0df5