Related papers: Finite element approximation of the Hardy constant

We study the behavior of the smallest possible constants $d(a,b)$ and $d_n$ in Hardy inequalities $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^p\,dx\leq d(a,b)\,\int_a^b [f(x)]^p dx $$ and $$…

Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions. Chang and Nakshatrala enforce…

The best known upper estimates for the constants of the Hardy--Littlewood inequality for $m$-linear forms on $\ell_{p}$ spaces are of the form $\left(\sqrt{2}\right) ^{m-1}.$ We present better estimates which depend on $p$ and $m$. An…

We propose and analyze an unfitted finite element method for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element…

We propose an optimal approximation formula for analytic functions that are defined on a complex region containing the real interval $(-1,1)$ and possibly have algebraic singularities at the endpoints of the interval. As a space of such…

We show that the rate of convergence of solutions of finite-difference approximations for uniformly elliptic Bellman's equations is of order at least $h^{2/3}$, where $h$ is the mesh size. The equations are considered in smooth bounded…

We show that the best constants appearing in the weak type (1,1) inequalities satisfied by the centered Hardy-Littlewood maximal function associated to some finite radial measures, such as the standard gaussian measure, grow exponentially…

We study finite element approximations of the nonhomogeneous Dirichlet problem for the fractional Laplacian. Our approach is based on weak imposition of the Dirichlet condition and incorporating a nonlocal analogous of the normal derivative…

Meshing of geometric domains having curved boundaries by affine simplices produces a polytopial approximation of those domains. The resulting error in the representation of the domain limits the accuracy of finite element methods based on…

Let $\Omega$ be an open connected cone in $\mathbb{R}^n$ with vertex at the origin. Assume that the operator $$P_\mu:=-\Delta-\frac{\mu}{\delta_\Omega^2(x)}$$ is {\em subcritical} in $\Omega$, where $\delta_\Omega$ is the distance function…

In this current work, we revisit the recent improvement of the discrete Hardy's inequality in one dimension and establish an extended improved discrete Hardy's inequality with its optimality. We also study one-dimensional discrete Copson's…

In this paper, we establish hardness and approximation results for various $L_p$-ball constrained homogeneous polynomial optimization problems, where $p \in [2,\infty]$. Specifically, we prove that for any given $d \ge 3$ and $p \in…

We present a review of results that have been obtained in the past twenty-five years concerning the $L^p$-Hardy inequality with distance to the boundary. We concentrate on results where the best Hardy constant is either computed exactly or…

The aim of this paper is to characterize in broad classes of means 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)…

We study the optimal domain for the Hardy operator considered with values in a rearrangement invariant space. In particular, this domain can be represented as the space of integrable functions with respect to a vector measure defined on a…

In this paper, we consider the finite element approximation for a parabolic problem on a smooth domain $\Omega \subset \mathbb{R}^N$ with the inhomogeneous Neumann boundary condition. We emphasize that the domain can be non-convex in…

A finite element approach for approximating the solution of a mathematical model for the response of a penetrable, bounded object (obstacle) to the excitation by an external electromagnetic field is presented and investigated. The model…

We present the best constant and the existence of extremal functions for an Improved Hardy-Sobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in $\mathbb{R}^N$. We also…

We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity…

We prove an optimal Hardy inequality for the fractional Laplacian on the half-space.