Eigenvalue bounds of mixed Steklov problems

We study bounds on the Riesz means of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalue problem on a bounded domain $\Omega$ in $\mathbb{R}^n$. The Steklov-Neumann eigenvalue problem is also called the sloshing problem. We obtain two-term asymptotically sharp lower bounds on the Riesz means of the sloshing problem and also provide an asymptotically sharp upper bound for the Riesz means of mixed Steklov-Dirichlet problem. The proof of our results for the sloshing problem uses the average variational principle and monotonicity of sloshing eigenvalues. In the case of Steklov-Dirichlet eigenvalue problem, the proof is based on a well-known bound on the Riesz means of the Dirichlet fractional Laplacian and an inequality between the Dirichlet and Navier fractional Laplacian. The two-term asymptotic results for the Riesz means of mixed Steklov eigenvalue problems are discussed in the appendix which in particular show the asymptotic sharpness of the bounds we obtain.Comment: An appendix by by F. Ferrulli and J. Lagac\'e is added; some changes in the introduction are mad

Inequalities between Dirichlet and Neumann eigenvalues on the Heisenberg group

We prove that for any domain in the Heisenberg group the (k+1)'th Neumann eigenvalue of the sub-Laplacian is strictly less than the k'th Dirichlet eigenvalue. As a byproduct we obtain similar inequalities for the Euclidean Laplacian with a homogeneous magnetic field.Comment: 10 page

Analysis of the time series in the space maser signals

We analyze the data of the observations of the radio sources frequently found in space. They are believed to be the sets of molecular condensations each of which works as a maser, so that the whole set produces a characteristic spectrum. It turns out that in some cases the intensity of one of the components of such spectrum corresponding to a single condensation changes periodically with a period of dozens of minutes or of hours.Comment: 5 pages, 6 figure