Related papers: A note on the boundary behaviour of the squeezing …

Let $D$ be a bounded domain in $\mathbb{C}^n$, $n\ge 1$. In this paper, we study two biholomorphic invariants on $D$, the Fridman invariant $e_D(z)$ and the squeezing function $s_D(z)$. More specifically, we study the following two…

In this paper, we study the geometry of bounded domains with piecewise smooth boundary. Specifically, we obtain the relationship between the squeezing function corresponding to polydisk and Levi flatness on bounded generic convex domains.…

We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains $\Omega \subset \mathbb{R}^n$. For a rotationally invariant Cheeger set $C$, the free boundary $\partial C \cap \Omega$ consists of pieces of…

Let $\,(M,g)\,$ be a $n$-dimensional Riemannian manifold and $\,\Omega\,$ be any compact connected domain in $\,M$. We study the problem of finding the {\em maxima} of the functional $\, {\mathcal E} (\Omega)\,$ (known as {\em torsional…

We present a method of obtaining a lower bound estimate of the curvatures of the Bergman metric without using the regularity of the kernel function on the boundary. As an application, we prove the existence of an uniform lower bound of the…

We study the behavior of the fractional convexity when the fractional parameter goes to 1. For any notion of convexity, the convex envelope of a datum prescribed on the boundary of a domain is defined as the largest possible convex function…

Given a Lipschitz domain $\Omega $ in ${\mathbb R} ^N $ and a nonnegative potential $V$ in $\Omega $ such that $V(x)\, d(x,\partial \Omega)^2$ is bounded in $\Omega $ we study the fine regularity of boundary points with respect to the…

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq1.$ For each $p>N$ we study the optimal function $s=s_{p}$ in the pointwise inequality \[ \left\vert v(x)\right\vert \leq s(x)\left\Vert \nabla v\right\Vert…

The aim of this study is to understand to what extent a 1-convex domain with Levi-flat boundary is capable of holomorphic functions with slow growth. This paper discusses a typical example of such domain, the space of all the geodesic…

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^2$ with Lipschitz boundary or a bounded convex domain in $\mathbb{C}^n$ and $\phi\in C(\overline{\Omega})$ such that $H_{\phi}$ is compact on $A^2(\Omega)$. Then $\phi\circ f$ is…

A theoretical analysis of the finite element method for a generalized Robin boundary value problem, which involves a second-order differential operator on the boundary, is presented. If $\Omega$ is a general smooth domain with a curved…

We show that for any bounded domain $\Omega\subset\Cp ^n$ of 1-type $2k $ which is locally convexifiable at $p\in b\Omega$, having a Stein neighborhood basis, there is a biholomorphic map $f:\bar{\Omega}\rightarrow \Cp ^n $ such that $f(p)$…

We construct a strictly pseudoconvex domain with smooth boundary whose squeezing function is not plurisubharmonic.

We consider a scalar valued elliptic partial differential equation on a sufficiently smooth domain $\Omega$, subject to a regularized Tresca friction-type boundary condition on a subset $\Gamma$ of $\partial \Omega$. The friction threshold,…

Let $\Omega$ be a Lipschitz bounded domain of $\mathbb{R}^N $, $N\geq2$. The fractional Cheeger constant $h_s (\Omega)$, $0<s<1$, is defined by \[h_s(\Omega)=\inf_{E\subset{\Omega}}\frac{P_s(E)}{|E|},\: \text{ where } \: P_s…

Let $\Omega \subset \mathbb{R}^2$ be a bounded, convex domain and let $u$ be the solution of $-\Delta u = 1$ vanishing on the boundary $\partial \Omega$. The estimate $$ \| \nabla u\|_{L^{\infty}(\Omega)} \leq c |\Omega|^{1/2}$$ is…

Let $(\Omega,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $\phi_{\lambda}$ with eigenvalue $\lambda^2$ and $u_{\lambda}:= \phi_{\lambda} |_{\partial \Omega}$ the associated…

We consider the functional $$I_\Omega(v) = \int_\Omega [f(|Dv|) - v] dx,$$ where $\Omega$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, G. Crasta [Cr1] has shown that if $I_\Omega$ admits a minimizer in…

The Diederich--Forn\ae ss index has been introduced since 1977 to classify bounded pseudoconvex domains. In this article, we derive several intrinsic, geometric conditions on boundary of domains for arbitrary indexes. Many results, in the…

We consider the first eigenvalue $\lambda_1(\Omega,\sigma)$ of the Laplacian with Robin boundary conditions on a compact Riemannian manifold $\Omega$ with smooth boundary, $\sigma\in\bf R$ being the Robin boundary parameter. When $\sigma>0$…