arXiv subjects

B. Siudeja

Publications and source records attributed to B. Siudeja.

6 recordsLinked to original sources

arXiv2010

Eigenvalue inequalities for mixed Steklov problems

We extend some classical inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian to the context of mixed Steklov--Dirichlet and Steklov--Neumann eigenvalue problems. The latter one is also known as the sloshing problem, and has been actively studied for more than a century due to its importance in hydrodynamics. The main results of the paper are applied to obtain certain geometric information about nodal sets of sloshing eigenfunctions. The key ideas of the proofs include domain monotonicity for eigenvalues of mixed Steklov problems, as well as an adaptation of Filonov's method developed originally to compare the Dirichlet and Neumann eigenvalues.

math.SP↗

arXiv2009

Minimizing Neumann fundamental tones of triangles: an optimal Poincare inequality

The first nonzero eigenvalue of the Neumann Laplacian is shown to be minimal for the degenerate acute isosceles triangle, among all triangles of given diameter. Hence an optimal Poincaré inequality for triangles is derived. The proof relies on symmetry of the Neumann fundamental mode for isosceles triangles with aperture less than $π/3$. Antisymmetry is proved for apertures greater than $π/3$.

math.SP↗

arXiv2009

Maximizing Neumann fundamental tones of triangles

We prove sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangular domains. The first nonzero Neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter, and hence among all triangles of given area. Similar results are proved for the harmonic and arithmetic means of the first two nonzero eigenvalues.

math.SP↗

arXiv2007

Neumann Heat kernel monotonicity

We prove that the diagonal of the transition probabilities for the d-dimensional Bessel processes on (0, 1], reflected at 1, which we denote by $p_R^N(t, r,r)$, is an increasing function of r for d>2 and that this is false for d=2.

math.PR↗

arXiv2007

Scattering length for stable processes

Let $α\in(0,2)$ and $X_t$ be a symmetric $α$-stable process. We define the scattering length $Γ(v)$ of the positive potential $v$ and prove several of its basic properties. We use the scattering length to findestimates for the first eigenvalue of the Schrödinger operator of the ``Neumann'' fractional Laplacian in a cube with potential $v$.

math.PR↗

arXiv2006

Sharp bounds for eigenvalues of triangles

We prove that the first eigenvalue of the Dirichlet Laplacian for a triangle in the plane is bounded above by $π^2 L^2\over 9A^2$, where $L$ is the perimeter and $A$ is the area of this triangle. We show that the \mbox{constant 9} is optimal and that the optimal constant for the lower bound of the same form is 16. This gives a positive answer to a conjecture made by P. Freitas.

math.SP↗