Related papers: Some remarks on the Cegrell's class $\mathcal{F}$

Let $\Omega$ be a bounded, weakly pseudoconvex domain in C^n, n > 1, with real-analytic boundary. A real-analytic submanifold $M \subset bd\Omega$ is called an analytic interpolation manifold if every real-analytic function on M extends to…

Consider a bounded strongly pseudo-convex domain $\Omega $ with a smooth boundary in $\mathbb{C}^n$. Let $\mathcal{T}$ be the Toeplitz algebra on the Bergman space $L^2_a(\Omega )$. That is, $\mathcal{T}$ is the $C^\ast $-algebra generated…

Let $f$ be a function on a bounded domain $\Omega \subseteq \mathbb{R}^n$ and $\delta$ be a positive function on $\Omega$ such that $B(x,\delta(x))\subseteq \Omega$. Let $\sigma(f)(x)$ be the average of $f$ over the ball $B(x,\delta(x))$.…

In this paper we study the class $\mathcal{U}$ of functions that are analytic in the open unit disk ${\mathbb D}=\{z:|z|<1\}$, normalized such that $f(0)=f'(0)-1=0$ and satisfy \[\left|\left [\frac{z}{f(z)} \right]^{2}f'(z)-1…

We consider the relaxation of polyconvex functionals with linear growth with respect to the strict convergence in the space of functions of bounded variation. These functionals appears as relaxation of $F(u,\Omega):=\int_\Omega f(\nabla…

In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in \cite{HS1} to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace…

Let $\Omega$ be a domain in a smooth complete Finsler manifold, and let $G$ be the largest open subset of $\Omega$ such that for every $x$ in $G$ there is a unique closest point from $\partial \Omega$ to $x$ (measured in the Finsler…

We consider the Bernoulli one-phase free boundary problem in a domain $\Omega$ and show that the free boundary $F$ is $C^{1,1/2}$ regular in a neighborhood of the fixed boundary $\partial \Omega$. We achieve this by relating the behavior of…

Let $\Omega \subset \mathbb{C}^n$ be a bounded domain and let $\mathcal{A} \subset \mathcal{C}(\bar{\Omega})$ be a uniform algebra generated by a set $F$ of holomorphic and pluriharmonic functions. Under natural assumptions on $\Omega$ and…

We study extremal properties of the function $$ F(x) := \min\{k\|x\|^{1-1/k}\colon k\ge 1\},\ x\in[0,1], $$ where $\|x\|=\min\{x,1-x\}$. In particular, we show that $F$ is the pointwise largest function of the class of all real-valued…

We prove that for every $n \ge 2$, there exists a pseudoconvex domain $\Omega \subset \mathbb{C}^n$ such that $\mathfrak{c}^0(\Omega) \subsetneq \mathfrak{c}^1(\Omega)$, where $\mathfrak{c}^k(\Omega)$ denotes the core of $\Omega$ with…

We prove bounds for the covering numbers of classes of convex functions and convex sets in Euclidean space. Previous results require the underlying convex functions or sets to be uniformly bounded. We relax this assumption and replace it…

In this paper, we introduce a subclass of close-to-convex functions defined in the open unit disk. We obtain the inclusion relationships, coefficient estimates and Fekete-Szego inequality. The results presented here would provide extension…

Let \Omega be a bounded, weakly convex domain in C^n, n>1, having real-analytic boundary. A(\Omega) is the algebra of all functions holomorphic in \Omega and continuous upto the boundary. A submanifold M\subset \partial\Omega is said to be…

We consider functionals of the form $$\mathcal{F}(u):=\int_\Omega\!F(x,u,\nabla u)\,\mathrm{d} x,$$ where $\Omega\subseteq\mathbb{R}^n$ is open and bounded. The integrand $F\colon\Omega\times\mathbb{R}^N\times\mathbb{R}^{N\times…

We introduce a wide subclass ${\cal F}(X,\omega)$ of quasi-plurisubharmonic functions in a compact K\"ahler manifold, on which the complex Monge-Amp\`ere operator is well-defined and the convergence theorem is valid. We also prove that…

Let $\Omega$ be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in $\Omega$. We study metric properties of this space using Mabuchi geodesics and establish regularity properties of the…

Let $\Omega\subset \mathbb C^n$ be a bounded domain, and let $f$ be a real-valued function defined on the whole topological boundary $\partial \Omega$. The aim of this paper is to find a characterization of the functions $f$ which can be…

For $\Omega$ varying among open bounded sets in ${\mathbb R} ^n$, we consider shape functionals $J (\Omega)$ defined as the infimum over a Sobolev space of an integral energy of the kind $\int _\Omega[ f (\nabla u) + g (u) ]$, under…

Let $\Omega$ be a domain in $R^n$, and let $N=3\cdot 2^{n-1}$. We prove that the trace of the space $C^2(\Omega)$ to the boundary of $\Omega$ has the following finiteness property: A function $f:\partial\Omega\to R$ is the trace to the…