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

Let $\Omega$ be a bounded strictly pseudoconvex domain of $\mathbb{C}^n$. We solve degenerate complex Monge-Amp\`ere equations of the form $(\omega + dd^c \varphi)^n = \mu$ in the generalized Cegrell classes $\mathcal{K}(\Omega,\omega,H)$,…

A sufficient condition for $\bar{\partial}$ to have closed range is given for pseudoconvex, possibly unbounded domains in $\mathbb{C}^n$.

The aim of this paper is twofold. First, we cut off a part of a convex surface by a plane near a ridge point and characterize the limiting behavior of the surface measure in $S^2$ induced by this part of surface when the plane approaches…

Let ($M$, $\Omega$) be a smooth symplectic manifold and $f:M\rightarrow M$ be a symplectic diffeomorphism of class $C^l$ ($l\geq 3$). Let $N$ be a compact submanifold of $M$ which is boundaryless and normally hyperbolic for $f$. We suppose…

For analytic functions f(z) in the closed unit disk \bar{U}, two boundary points z_1 and z_2 such that \alpha = (f'(z_1)+f'(z_2))/2 in f'(U) are considered. The object of the present paper is to discuss some interesting conditions for f(z)…

The purpose of this paper is to study the lower semicontinuity with respect to the strong $L^1$-convergence, of some integral functionals defined in the space SBD of special functions with bounded deformation. Precisely, let $U$ be a…

We present a constructive lower bound of $\Omega(1/\sqrt{\varepsilon})$ for Frank-Wolfe (FW) when both the objective and the constraint set are smooth and strongly convex, showing that the known uniform $\mathcal{O}(1/\sqrt{\varepsilon})$…

Let $\Omega\subset\mathbb{C}^m$ be a bounded pseudoconvex domain with smooth boundary. For each $k\in\mathbb{N}$, we give a sufficient condition to estimate the $\bar\partial$-Neumann operator in the Sobolev space $W^k(\Omega)$. The key…

For $\alpha > -1$ and $\beta >0, $ let $\mathcal{B}_{\mathcal{H}}^0(\alpha, \beta)$ denote the class of sense preserving harmonic mappings $f=h+\overline{g}$ in the open unit disk $\mathbb{D}$ satisfying $|zh''(z)+\alpha(h'(z)-1)|\leq…

We show that every strictly pseudoconvex domain $\Omega$ with smooth boundary in a complex manifold $\mathcal{M}$ admits a global defining function, i.e., a smooth plurisubharmonic function $\varphi \colon U \to \mathbb R$ defined on an…

We consider a new subclass $\widetilde{\mathcal{K}}_u$ of close-to-convex functions in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$. For this class, we obtain sharp estimates of the Fekete-Szeg\"{o} problem, growth and distortion…

Let $\mathcal{A}_n$ be the class of analytic functions $f(z)$ of the form $f(z)=z+\sum_{k=n+1}^\infty a_kz^k,n\in\mathbb{N}$ and let \begin{align*} \Omega_n:=\left\{f\in\mathcal{A}_n:\left|zf'(z)-f(z)\right|<\frac{1}{2},\;…

In this article we consider functions $f$ meromorphic in the unit disk. We give an elementary proof for a condition that is sufficient for the univalence of such functions. This condition simplifies and generalizes known conditions. We…

Let ${\mathcal F}_I=\{f:I \to I| f(x)= (Ax+B)/(Cx+D); AD-BC \neq 0 \}$, where $I$ is an interval. For $x\in I$, let ${\Omega}_x$ be the orbit of $x$ under the action of the semigroup of functions generated by $f,g \in {\mathcal F}_I$. Our…

We investigate general semilinear (obstacle-like) problems of the form $\Delta u = f(u)$, where $f(u)$ has a singularity/jump at $\{u=0\}$ giving rise to a free boundary. Unlike many works on such equations where $f$ is approximately…

We establish a mixed observability inequality for a class of degenerate hyperbolic equations on the cylindrical domain $\Omega = \mathbb{T} \times (0,1)$ with mixed Neumann Dirichlet boundary conditions. The degeneracy acts only in the…

In this paper we study a partially overdetermined mixed boundary value problem for domains $\Omega$ contained in an unbounded set $\mathcal C$. We introduce the notion of Cheeger set relative to $\mathcal C$ and show that if a domain…

Assume that $p > 1$ and $p - 1 \le \alpha \le p$ are real numbers and $\Omega$ is a non-empty open subset of ${\mathbb R}^n$, $n \ge 2$. We consider the inequality $$ {\rm div} \, A (x, D u) + b (x) |D u|^\alpha \ge 0, $$ where $D =…

In this paper we use the dynamical methods to establish the existence of nontrivial solution for a class of nonlocal problem of the type $$ \left\{\begin{array}{l} -a\left(x,\int_{\Omega}g(u)\,dx \right)\Delta u =f(u), \quad x \in \Omega \\…

There are many results for sufficient conditions of functions f(z) which are analytic in the open unit disc U to be starlike and convex in U. The object of the present paper is to derive some interesting sufficient conditions for f(z) to be…