Related papers: Weinstock inequality in hyperbolic space

This paper investigates the first Dirichlet eigenvalue for the $p$-Laplacian in Riemannian manifolds. Firstly, we establish a lower bound for this eigenvalue under the condition that the domain includes a specific function which fulfills…

In this paper, we first construct a sequence of hyperbolic surfaces with connected geodesic boundary such that the first normalized Steklov eigenvalue $\tilde{\sigma}_1$ tends to infinity. We then prove that as $g\rightarrow \infty$, a…

Motivated by an example of Shih, we compute the fundamental gap of a family of convex domains in the hyperbolic plane $\mathbb H^2$, showing that for some of them $\lambda_2 - \lambda_1 < \frac{3\pi^2}{D^2}$, where $D$ is the diameter of…

The classical Minkowski inequality implies that the volume of a bounded convex domain is controlled from above by the integral of the mean curvature of its boundary. In this note, we establish an analogous inequality without the convexity…

We derive the equation of a free vibrating thin plate whose mass is concentrated at the boundary, namely a Steklov problem for the biharmonic operator. We provide Hadamard-type formulas for the shape derivatives of the corresponding…

Let $M^n$ be a closed convex hypersurface lying in a convex ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reilly's inequality via sectional curvature upper bound of $B(p,R)$, 1st eigenvalue and…

In this paper, we address the problem of maximizing the Steklov eigenvalues with a diameter constraint. We provide an estimate of the Steklov eigenvalues for a convex domain in terms of its diameter and volume and we show the existence of…

We prove that any Loewner PDE in a complete hyperbolic starlike domain of $\C^N$ (in particular in bounded convex domains) admits an essentially unique univalent solution with values in $\C^N$.

We investigate the qualitative properties of the weak solutions to the boundary value problems for the hyperbolic fourth-order linear equations with constant coefficients in the plane bounded domain convex with respect to characteristics.…

We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains $\Omega\subset\mathbb R^2$. With the help of these estimates we obtain asymptotically sharp inequalities of ratios of eigenvalues in…

We provide a resolution of the Heesch problem for homogeneous (also known as semi-regular) tilings, and as a corollary, for tilings by convex monotiles in the hyperbolic plane. We also provide the first known example of weakly aperiodic…

We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume…

In Euclidean and Hyperbolic space, and the hemisphere in $S^n$, geodesic balls maximize the gap $\lambda_2 - \lambda_1$ of Dirichlet eigenvalues, amoung domains with fixed $\lambda_1$. We prove an upper bound on $\lambda_2 - \lambda_1$ for…

We discuss isoperimetric inequalities for the magnetic Laplacian on bounded domains of $\mathbb R^2$ endowed with an Aharonov-Bohm potential. When the flux of the potential around the pole is not an integer, the lowest eigenvalue for the…

We prove that the cylindrical capacity of a dynamically convex domain in $\mathbb{R}^4$ agrees with the least symplectic area of a disk-like global surface of section of the Reeb flow on the boundary of the domain. Moreover, we prove the…

For a parabolically convex domain $M\subseteq \mathbb{H}^n$, $n\ge 3$, we prove that if $f:(N,\bar g)\to (M,g)$ has nonzero degree, where $N$ is spin with scalar curvature $R_N\ge -n(n-1)$, and if $f|_{\partial N}$ does not increase the…

We establish a lower bound on the total mass of the time slices of (n + 1)-dimensional asymptotically flat standard static spacetimes under the timelike convergence condition. The inequality can be viewed equivalently as a Minkowski-type…

We consider the first positive Steklov eigenvalue on planar domains. First, we provide an example of a planar domain for which a first eigenfunction has a closed nodal line. Second, we establish a lower bound for the first positive…

We show there exists a nontrivial $H^1_0$ solution to the steady Stokes equation on the 2D exterior domain in the hyperbolic plane. Hence we show there is no Stokes paradox in the hyperbolic setting. We also show the existence of a…

In [SWW16, HW17] it is shown that the difference of the first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter $D$ of sphere $\mathbb S^n$ is $\geq 3 \frac{\pi^2}{D^2}$ when $n \geq 3$. We…