Calculus of Variations and Geometric Measure Theory

H. M. Nguyen - Q. H. Nguyen

Discreteness of interior transmission eigenvalues revisited

created by nguyen on 23 May 2017


Published Paper

Inserted: 23 may 2017
Last Updated: 23 may 2017

Journal: Calculus of Variations and Partial Differential Equations
Year: 2017

ArXiv: 1610.06961 PDF


This paper is devoted to the discreteness of the transmission eigenvalue problems. It is known that this problem is not self-adjoint and a priori estimates are non-standard and do not hold in general. Two approaches are used. The first one is based on the multiplier technique and the second one is based on the Fourier analysis. The key point of the analysis is to establish the compactness and the uniqueness for Cauchy problems under various conditions. Using these approaches, we are able to rediscover quite a few known discreteness results in the literature and obtain various new results for which only the information near the boundary are required and there might be no contrast of the coefficients on the boundary.