Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.08028

Estimates of the Poincare constants for bounded domains in Euclidian space R^n

Bernd Rummler·Gudrun Thäter
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study bounded, simply connected, convex, and C2-smooth domains in the space R^n for finite n greater than 1. For such domains we use the diameter to find balls with the property that they are included in this ball, while the diameter of the ball is chosen to be minimal. With a simple transformation we are able to map our domain onto a domain inside the open unit ball. We derive estimates for the Poincare constants of scalar and vector functions on those domains. The main ingredient is the first eigenvalue of the Laplacian or the Stokes operator with vanishing Dirichlet traces, respectively (we use results of [21]). Additionally we apply the relation of the first eigenvalues and one eigenfunction of the (scalar) Laplace and the Stokes operator cf. [21]. Our estimates for the Poincare constants are new. The dimension n of the space has significant effect on the quality of our estimates. This is illustrated with the help of an example.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bernd Rummler, Gudrun Thäter. 2026-10-06. Estimates of the Poincare constants for bounded domains in Euclidian space R^n. https://arxiv.org/abs/2610.08028

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Global-in-time Well-posedness of Classical Solutions to the Vacuum Free Boundary Problem for the 1-D Viscous Saint-Venant System with Large Data

We establishes the global existence and uniqueness of classical solutions to the vacuum free boundary problem for the 1-D viscous Saint-Venant system with a general class of large initial data. Since the fluid depth $ρ$ vanishes on the moving boundary, the momentum equations degenerate in both time evolution and spatial dissipation, potentially causing singularities in velocity derivatives and making classical solutions difficult to construct. By exploiting the intrinsic degenerate-singular structures, we identify admissible initial depth profiles for which $ρ_0^α$ belongs to $H^3$ and vanishes as the distance to the moving boundary, with $\frac{1}{3}<α\le 1$. In particular, for $α=1$, it satisfies the physical vacuum boundary condition but violates the BD entropy condition. First, we introduce new weighted nonlinear energy functionals involving lower- and higher-order derivatives, based on the balance between pressure and spatial dissipation. Second, we derive new global weighted $L^p$ estimates, $2\le p\le\infty$, for the effective velocity $v=u+(\logρ)_y$, where $y$ is the Eulerian coordinate, using the transport properties of its evolution equation. These estimates differ from those of Bresch-Desjardins (Comm. Math. Phys. 238 (2003), 211-223) and Kanel (Differ. Uravn. 4 (1968), 721-734) and are crucial for $α=1$, where the BD entropy condition fails. Finally, we establish weighted energy estimates for first-order velocity derivatives with weights involving powers of $ρ_0$ and $η_x$, where $x$ is the Lagrangian coordinate and $η$ the flow map. These estimates give an upper bound for $η_x$ on every finite time interval without any smallness assumption on the initial data. Further singular or degenerate weighted energy estimates yield the desired global regularity.

math.AP↗
arXiv2026

Desingularization of vortex sheets for the 2D Euler equations

We show how to regularize vortex sheets by means of smooth, compactly supported vorticities that asymptotically evolve according to the Birkhoff-Rott vortex sheet dynamics. More precisely, consider a vortex sheet initial datum $ω^0_{\mathrm{sing}}$, which is a signed Radon measure supported on a closed curve. We construct a family of initial vorticities $ω^0_\varepsilon \in C^\infty_c(\mathbb{R}^2)$ converging to $ω^0_{\mathrm{sing}}$ distributionally as $\varepsilon \to 0^+$, and show that the corresponding solutions $ω_\varepsilon(x,t)$ to the 2D incompressible Euler equations converge to the measure defined by the Birkhoff-Rott system with initial datum $ω^0_{\mathrm{sing}}$. The regularization relies on a layer construction designed to exploit the key observation that the Kelvin-Helmholtz instability has a strongly anisotropic effect: while vorticities must be analytic in the "tangential" direction, the way layers can be arranged in the "normal" direction is essentially arbitrary.

math.AP↗
arXiv2026

Global-in-time convergence from bipolar Euler-Poisson equations to unipolar ones

In this paper, the Cauchy problem for the multi-dimensional (M-D) bipolar Euler-Poisson equations with far field vacuum is considered. Based on physical observations and some elaborate analysis of this system's intrinsic structures, for a class of smooth initial data that are of small scaled density but possibly large mean velocity, we give one rigorous global-in-time convergence proof for regular solutions from M-D bipolar Euler-Poisson equations to M-D unipolar Euler-Poisson equations through the vanishing electron-to-ion mass-ratio limit. Here the initial scaled density is required to vanish in the far field, and the spectrum of the Jacobian matrix of the initial mean velocity stays uniformly away from the nonpositive real axis. In order to deal with singular limits of this kind, the global-in-time uniform tame estimates of regular solutions to M-D bipolar Euler-Poisson equations with respect to the mass ratio are established, based on which the corresponding error estimates in smooth function spaces between the two systems considered are given. To this end, we establish global a priori estimates for solutions with compactly supported initial densities, uniformly in the mass ratio and the initial support radii. Compactness then yields global regular solutions for general initial densities.

math.AP↗