Related papers: On some extremal problems in spaces of harmonic fu…
For a real constant $b,$ we give sharp estimates of $\log|f(z)/z|+b\arg[f(z)/z]$ for subclasses of normalized univalent functions $f$ on the unit disk.
In this short note, we obtain error estimates for Riemann sums of some singular functions.
We study the distance function to the boundary, Finsler geometry and the singular set of viscosity solutions of some Hamilton-Jacobi equations.
We consider the set of extremal points of the generalized unit ball induced by gradient total variation seminorms for vector-valued functions on bounded Euclidean domains. These are central to the understanding of sparse solutions and…
The aim of this work is to derive new explicit solutions to the $\infty$-Laplace equation, the fundamental PDE arising in Calculus of Variations in the space $L^\infty$. These solutions obey certain symmetry conditions and are derived in…
In this paper we present some new results on the existence of solutions of generalized variational inequalities in real reflexive Banach spaces with Fr\'echet differentiable norms. Moreover, we also give some theorems about the structure of…
We study harmonic functions for the Laplace-Beltrami operator on the real hyperbolic ball. We obtain necessary and sufficient conditions for this functions and their normal derivatives to have a boundary distribution.In doing so, we put…
\begin{abstract} This paper deals with an extremal problem for bounded harmonic functions in the unit ball $\mathbb{B}^n.$ We solve the Khavinson conjecture in $\mathbb{R}^3,$ an intriguing open question since 1992 posed by D. Khavinson,…
We prove the existence of one or more solutions to a singularly perturbed elliptic problema with two potential functions.
We give a survey about the Runge approximation problem for a holomorphic function defined on the unit ball of a complex Banach space.
We survey the variants of Erd\H{o}s' distinct distances problem and the current best bounds for each of those.
In this work, we investigate some connections between exact differential equations and harmonic functions and in particular, we obtain necessary and sufficient conditions for which exact equations admit harmonic solutions. As an…
We investigate the properties of a modulus of a foliation on a Riemannian manifold. We give necessary and sufficient conditions for the existence of an extremal function and state some of its properties. We obtain the integral formula…
The solutions of a kind of second-order homogeneous partial differential equation are called (real kernel) alpha-harmonic functions. The alpha-harmonic functions and their first-order partial derivative functions on unit disk are estimated…
We prove the stronger version of Harnack's inequality for positive harmonic functions defined on the unit disc.
We show that nontrivial solutions to higher and fractional order equations with certain nonlinearity are radially symmetric and nonincreasing on geodesic balls in the hyperbolic space $\mathbb{H}^n$ as well as on the entire space…
We study lower bounds for the norm of the product of polynomials and their applications to the so called \emph{plank problem.} We are particularly interested in polynomials on finite dimensional Banach spaces, in which case our results…
This paper investigates the existence, nonexistence, and qualitative properties of p-harmonic functions in the upper half-space $\mathbb{R}^N_+ \, (N \geq 3)$ satisfying nonlinear boundary conditions for $1<p<N$. Moreover, the symmetry of…
Our first aim of this article is to establish several new versions of refined Bohr inequalities for bounded analytic functions in the unit disk involving Schwarz functions. Secondly, %as applications of these results, we obtain several new…
In this article, we establish radial symmetry for positive weak solutions of a class of mixed local-nonlocal equations with possibly singular nonlinearity via the moving plane method. Furthermore, we provide a quantitative version of…