Related papers: An Example on $s$-H-Convexity in $\mathbb{C}^2$

Let $\Omega\subset\mathbb{C}$ be an open set. We show that $\overline{\partial}$ has closed range in $L^{2}(\Omega)$ if and only if the Poincar\'e-Dirichlet inequality holds. Moreover, we give necessary and sufficient potential-theoretic…

In this paper, we prove that the first eigenfunction of the Laplacian for a horo-convex domain $\Omega\subset\mathbb H^n$ is super log-concave when $\text{diam}(\Omega)$ is not large. Our result is optimal in the sense that there are…

We provide a complete characterisation of extreme points of the space of sofic representations. We also show that the restriction map $Sof(G,P^{\omega})$ to $Sof(H,P^{\omega})$, where $H\subset G$ is not always surjective. The first part of…

It is known that, if $\Omega$ $\subset$ C is a convex set containing the numerical range of an operator A, then $\Omega$ is a C $\Omega$ -spectral set for A with C $\Omega$ $\le$ 1+ $\sqrt$ 2. We improve this estimate in unbounded cases.

In this article, we consider a bounded pseudoconvex domain in ${\bf C}^2$ satifying: (a) it admits a proper holomorphic mapping $f$ onto the unit ball $B^2$, and (b) it is simply connected and has a real analytic boundary. According to…

Let $\Omega \subset R^n$, $n \geq 3$, be a fixed smooth bounded domain, and let $\gamma$ be a smooth conductivity in $\overline{\Omega}$. Consider a non-zero frequency $\lambda_0$ which does not belong to the Dirichlet spectrum of $L_\gamma…

We derive a decomposition result for regular, two-dimensional domains into John domains with uniform constants. We prove that for every simply connected domain $\Omega \subset {\Bbb R}^2$ with $C^1$-boundary there is a corresponding…

An oriented link L in a 3-sphere S in complex 2-space is a C-boundary if it bounds a piece of algebraic curve in the 4-ball bounded by S. Using Kronheimer and Mrowka's proof of the Thom Conjecture, we construct many oriented knots which are…

Let $\Omega$ be an open convex domain of the complex plane. We study constants K such that $\Omega$ is K-spectral or complete K-spectral for each continuous linear Hilbert space operator with numerical range included in $\Omega$. Several…

We prove that if two $C^{1,1}(\Omega)$ solutions of the second boundary value problem for the generated Jacobian equation intersect in $\Omega$ then they are the same solution. In addition we extend this result to $C^{2}(\overline{\Omega})$…

We give, in dimensions three or greater, an example of a bounded, pseudoconvex, circular domain in complex space with smooth real analytic boundary and non-compact automorphism group which is not biholomorphically equivalent to any…

We get a new multiplicity result for gradient systems. Here is a very particular corollary: Let $\Omega\subset {\bf R}^n$ ($n\geq 2$) be a smooth bounded domain and let $\Phi:{\bf R}^2\to {\bf R}$ be a $C^1$ function, with $\Phi(0,0)=0$,…

In this new version, we give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in $\mathbb{C}^n, n\geq 2,$ is K\"ahler-Einstein if and…

We study symmetry and quantitative approximate symmetry for an overdetermined problem involving the fractional torsion problem in a bounded domain $\Omega \subset \mathbb R^n$. More precisely, we prove that if the fractional torsion…

We reduce the problem of constructing a linear solution operator to the $\bar{\partial}$-equation on smoothly bounded weakly pseudoconvex domains, $\Omega$, in $\mathbb{C}^2$ to the problem of the boundary $\bar{\partial}_b$-equation. We…

Let $\Omega$ be a compact convex domain in the plane. We prove that $L^2(\Omega)$ has an orthogonal basis of exponentials if and only if $\Omega$ tiles the plane by translation.

Let $\Omega$ be an open set in $\mathbb{R}^n$ with $C^1$-boundary and $\Sigma$ be the skeleton of $\Omega$, which consists of points where the distance function to $\partial\Omega$ is not differentiable. This paper characterizes the cut…

We give an example of a bounded, pseudoconvex, circular domain in ${\mathbb C}^3$ with smooth, real-analytic boundary and non-compact automorphism group, which is not biholomorphically equivalent to any Reinhardt domain.

In this paper we prove: if a bounded domain with $C^2$ boundary covers a manifold which has finite volume with respect to either the Bergman volume, the K\"ahler-Einstein volume, or the Kobayashi-Eisenman volume, then the domain is…

The Hardy constant of a simply connected domain $\Omega\subset\mathbf{R}^2$ is the best constant for the inequality \[ \int_{\Omega}|\nabla u|^2dx \geq c\int_{\Omega} \frac{u^2}{{\rm dist}(x,\partial\Omega)^2}\, dx \; , \;\;\quad u\in…