2019/07/26 by Mateusz Kwaśnicki, Kwaśnicki, Mateusz
Computer Science · Mathematics · #35R11 (35J25 35J70 35S30 47G20 60J60 60J75) #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math.AP #msc:35J70 #msc:35R11 #msc:35S30 #msc:47G20 #msc:60J60
paper · pdf · doi:10.48550/arxiv.1907.11444
40 pages
openalex publication_date 2019/07/26 · arxiv created 2019/08/01 · arxiv updated 2019/08/02 · openalex created_date 2022/08/24 · openalex updated_date 2026/07/28
We identify a class of non-local integro-differential operators K in ℝ with Dirichlet-to-Neumann maps in the half-plane ℝ × (0, ∞) for appropriate elliptic operators L. More precisely, we prove a bijective correspondence between Lévy operators K with non-local kernels of the form ν(y - x), where ν(x) and ν(-x) are completely monotone functions on (0, ∞), and elliptic operators L = a(y) ∂xx + 2 b(y) ∂x y + ∂yy. This extends a number of previous results in the area, where symmetric operators have been studied: the classical identification of the Dirichlet-to-Neumann operator for the Laplace operator in ℝ × (0, ∞) with -√-∂xx, the square root of one-dimensional Laplace operator; the Caffarelli--Silvestre identification of the Dirichlet-to-Neumann operator for ∇ ⋅ (y1 - α ∇) with (-∂xx)α/2 for α∈ (0, 2); and the identification of Dirichlet-to-Neumann maps for operators a(y) ∂xx + ∂yy with complete Bernstein functions of -∂xx due to Mucha and the author. Our results rely on recent extension of Krein's spectral theory of strings by Eckhardt and Kostenko.