arXiv subjects

Bernd Rummler

Publications and source records attributed to Bernd Rummler.

5 recordsLinked to original sources

arXiv2026

Explicit formulas for gradients and the divergence in n-dimensional spherical coordinates

We use the Laplacian in n-dimensional spherical coordinates (n>1) to write the divergence of a vector field defined on radially symmetric domains in the context of vector calculus. We apply straightforward equations of vector calculus with the nabla operator and the transformation matrices from Cartesian to spherical polar coordinates. One needs the divergence of a vector field e.g. to prove that vector fields are eigenfunctions of the Stokes operator on n-dimensional annuli and balls. Our divergence formula in partial derivatives in n-dimensional spherical polar coordinates is an important step in a future verification of further Stokes eigenfunctions on those domains.

math.AP↗

arXiv2026

Exact Poincare Constants in n-dimensional Annuli

We study $n$-dimensional annuli for $n\,\in\,\{2,\dots,N\}$ with $N\,<\,\infty$. We choose a non-dimensional setting such that for any fixed $n $ and given number ${\cal A}>0$ the annuli ${\Omega}_{(n),\cal A}$ are defined as space between two concentrical balls with radii ${\cal A}/2$ and ${\cal A}/2 +1$ in ${ R}^{n}$. For these geometries we provide calculated (precise) Poincar\'e constants. These depend on ${\cal A}$ and the dimension $n$. Additionally we find a direct match of the Poincar\'e constants for solenoidal vector fields in ${R}^{n}$ and the Poincar\'e constants for scalar functions in ${ R}^{n+2}$ (all with vanishing Dirichlet traces). This is based on the relation of the first eigenvalues and one eigenfunction of the (scalar) Laplace and the Stokes operator. In addition we consider the limit ${\cal A}\,\to\,0$. In this context problems in domains ${\Omega}_{(n),\sigma}^{*}$ are investigated. These domains enable us to use the Green's function of the Laplacian with vanishing Dirichlet traces to show that the first eigenvalue here tends to the first eigenvalue of the corresponding problem on the open unit ball in ${ R}^{n}$. On the other hand, we take advantage of the so-called small-gap limit for ${\cal A}\to\infty$.

math.AP↗

arXiv2025

Exact Poincar\'e Constants in three-dimensional Annuli

We study 3d-annuli. In our non-dimensional setting each annulus ${\Omega}_{\cal A}$ is defined via two concentrical balls with radii ${\cal A}/2$ and ${\cal A}/2 +1$. For these geometries we provide the exact value for the Poincar\'e constants for scalar functions and calculate precise Poincar\'e constants for solenoidal vector fields (in both cases with vanishing Dirichlet traces on the boundary). For this we use the first eigenvalues of the scalar Laplacian and the Stokes operator, respectively. Additionally, corresponding problems in domains ${\Omega}_{\sigma}^{*}$, the 3d-annuli are investigated - for comparison but also to provide limits for ${\cal A}\,\to\,0$. In particular, the Green's function of the Laplacian on ${\Omega}_{\sigma}^{*}$ with vanishing Dirichlet traces on $\partial {\Omega}_{\sigma}^{*}$ is used to show that for ${\sigma}\,\to\,0$ the first eigenvalue here tends to the first eigenvalue of the corresponding problem on the open unit ball. On the other hand, we take advantage of the so-called small-gap limit for ${\cal A}\to\infty$.

math.AP↗

arXiv2025

Natural convection in the horizontal annulus: critical Rayleigh number for the steady problem

For the 2D Oberbeck-Boussinesq system in an annulus we are looking for the critical Rayleigh number for which the (nonzero) basic flow loses stability. For this we consider the corresponding Euler-Lagrange equations and construct a precise functional analytical frame for the Laplace- and the Stokes problem as well as the Bilaplacian operator in this domain. With this frame and the right set of basis functions it is then possible to construct and apply a numerical scheme providing the critical Rayleigh number.

math.AP↗

arXiv2024

The Stokes Eigenvalue Problem on balls and annuli in three dimensions: Solutions with Poloidal and Toroidal Fields

We consider the Stokes eigenvalue problem in open balls and open annuli in R3 with homogeneous Dirichlet boundary conditions. Using the frame of toroidal and poloidal fields we construct the othogonal decomposition of the Stokes eigenvalue problem in problems for toroidal and poloidal eigenfunctions. This provides the proof of the completeness of a system of explicitly calculated Stokes eigenfunctions given by one of the authors in 1999, [14].

math.AP↗