相关论文: Monotonicity, continuity and differentiability res…
The aim of this paper is to characterize the so-called Hardy means, i.e., those means $M\colon\bigcup_{n=1}^\infty \mathbb{R}_+^n\to\mathbb{R}_+$ that satisfy the inequality $$ \sum_{n=1}^\infty M(x_1,\dots,x_n) \le C\sum_{n=1}^\infty x_n…
We derive an improved Poincar\'e inequality in connection with the Babu\v{s}ka-Aziz and Friedrichs-Velte inequalities for differential forms by estimating the domain specific optimal constants figuring in the respective inequalities with…
In this paper, we study derivatives of powers of Euclidean norm. We prove their H\"older continuity and establish explicit expressions for the corresponding constants. We show that these constants are optimal for odd derivatives and at most…
To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is…
This paper deals with the principal eigenvalue of discrete $p$-Laplacian on the set of nonnegative integers. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequalities. The main goal is the quantitative…
The aim of this paper is to propose an abstract construction of spaces which keep the main properties of the (already known) Hardy spaces H^1. We construct spaces through an atomic (or molecular) decomposition. We prove some results about…
We derive a simple criterion that ensures uniqueness, Lipschitz stability and global convergence of Newton's method for the finite dimensional zero-finding problem of a continuously differentiable, pointwise convex and monotonic function.…
We prove general theorems for isoperimetric problems on lattices of the form ${\mathbb{Z}}^{k} \times {\mathbb{N}}^{d}$ which state that the perimeter of the optimal set is a monotonically increasing function of the volume under certain…
When studying the weighted Hardy-Rellich inequality in $L^2$ with the full gradient replaced by the radial derivative the best constant becomes trivially larger or equal than in the first situation. Our contribution is to determine the new…
We prove that in variable exponent spaces $L^{p(\cdot)}(\Omega)$, where $p(\cdot)$ satisfies the log-condition and $\Omega$ is a bounded domain in $\mathbf R^n$ with the property that $\mathbf R^n \backslash \bar{\Omega}$ has the cone…
We compute the explicit sharp constants of Hardy inequalities in the cone $\mathbb{R}_{k_+}^{n}:=\mathbb{R}^{n-k}\times (\mathbb{R}_{+})^{k}=\{(x_{1},...,x_{n})|x_{n-k+1}>0,...,x_{n}>0\}$ with $1\leq k\leq n$. Furthermore, the spherical…
We prove geometric $L^p$ versions of Hardy's inequality for the sub-elliptic Laplacian on convex domains $\Omega$ in the Heisenberg group $\mathbb{H}^n$, where convex is meant in the Euclidean sense. When $p=2$ and $\Omega$ is the…
We consider the series expansion of the $L^p$-Hardy inequality of \cite{BFT2}, in the particular case where the distance is taken from an interior point of a bounded domain in $\mathbb{R}^n$ and $1<p\neq n$. For $p<n$ we improve it by…
This paper is devoted to the study of $L_p$ Lyapunov-type inequalities for linear systems of equations with Neumann boundary conditions and for any constant $p \geq 1$. We consider ordinary and elliptic problems. The results obtained in the…
Motivated by a discrete inequality problem proposed by Duanyang Zhang as Problem 6 of the 2022 Spring NSMO, we prove a median version of Hardy's inequality. For a nonnegative function $f\in L^p(0,\infty)$, $p>1$, let $A(t)$ be the average…
In this paper, we establish discrete Hardy-Rellich inequalities on $\mathbb{N}$ with $\Delta^\frac{\ell}{2}$ and optimal constants, for any $\ell \geq 1$. As far as we are aware, these sharp inequalities are new for $\ell \geq 3$. Our…
We obtain conditions for the differentiability of weak solutions for a second-order uniformly elliptic equation in divergence form with a homogeneous co-normal boundary condition. The modulus of continuity for the coefficients is assumed to…
In the present paper we shall study a variational problem relating the weighted Hardy inequalities with sharp missing terms. As weights we treat non-doubling functions of the distance to the boundary of bounded domain.
The objective of this paper is to characterize harmonic Hardy spaces and a boundary behavior of harmonic functions on a smooth domain in real Euclidean space.
Hardy-Littlewood-Sobolev (HLS) Inequality fails in the "critical" case: \mu=n. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case: \mu=n and p=q, by limiting the inequality…