Related papers: Optimal Hardy inequalities in cones

\begin{abstract} We state the following weighted Hardy inequality \begin{equation*} c_{o, \mu}\int_{{\R}^N}\frac{\varphi^2 }{|x|^2}\, d\mu\le \int_{{\R}^N} |\nabla\varphi|^2 \, d\mu + K \int_{\R^N}\varphi^2 \, d\mu \quad \forall\, \varphi…

We study the following version of Hardy-type inequality on a domain $\Omega$ in a Riemannian manifold $(M,g)$: $$ \int{\Omega}|\nabla u|_g^p\rho^\alpha dV_g \geq \left(\frac{|p-1+\beta|}{p}\right)^p\int{\Omega}\frac{|u|^p|\nabla…

Let $\Omega\subset{\mathbb R}^2$ be a bounded domain on which Hardy's inequality holds. We prove that $[\exp(u^2)-1]/\delta^2\in L^1(\Omega)$ if $u\in H^1_0(\Omega)$, where $\delta$ denotes the distance to $\partial\Omega$. The…

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…

\begin{abstract} In this paper we state the following weighted Hardy type inequality for any functions $\varphi$ in a weighted Sobolev space and for weight functions $\mu$ of a quite general type \begin{equation*} c_{N,\mu}…

This paper establishes a bivariate Hardy-Sobolev inequality. Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be an open domain, $s \in (0,2)$, $\alpha > 1$, $\beta > 1$ with $\alpha + \beta = 2^*(s)$, and $\kappa \in \mathbb{R}$. For any…

In this expository paper, we consider the Hardy-Schr\"odinger operator $-\Delta -\gamma/|x|^2$ on a smooth domain \Omega of R^n with 0\in\bar{\Omega}, and describe how the location of the singularity 0, be it in the interior of \Omega or on…

The paper concentrates on the application of the following Hardy inequality \begin{equation*} \int_\Omega \ |\xi(x)|^p \omega_{1 }(x)dx\le \int_\Omega |\nabla \xi(x)|^p\omega_{2 }(x)dx, \end{equation*} to the proof of existence of weak…

In this paper we state the weighted Hardy inequality \begin{equation*} c\int_{{\mathbb R}^N}\sum_{i=1}^n \frac{\varphi^2 }{|x-a_i|^2}\, \mu(x)dx\le \int_{{\mathbb R}^N} |\nabla\varphi|^2 \, \mu(x)dx +k \int_{\mathbb{R}^N}\varphi^2 \,…

We prove some Hardy type inequalities related to quasilinear second order degenerate elliptic differential operators L_p(u):=-\nabla_L^*(\abs{\nabla_L u}^{p-2}\nabla_L u). If \phi is a positive weight such that -L_p\phi>= 0, then the Hardy…

\begin{abstract} In the paper we state conditions on potentials $V$ to get the improved Hardy inequality with weight \begin{equation*} \begin{split} c_{N,\mu}\int_{\R^N}\frac{\varphi^2}{|x|^2}\mu(x)dx&+ \int_{\R^N}V\,\varphi^2\mu(x)dx…

Let P be a linear, second order, elliptic operator satisfying a Hardy inequality with potential W (i.e. $P-W\geq0$) and best constant $\alpha$. We give conditions so that the spectrum of $W^{-1}P$ is $[\alpha,\infty)$. We apply this to…

We investigate the Hardy-Schr\"odinger operator $L_\gamma=-\Delta -\frac{\gamma}{|x|^2}$ on domains $\Omega\subset\rn$, whose boundary contain the singularity $0$. The situation is quite different from the well-studied case when $0$ is in…

Fix an integer $n\geq 2$, an exponent $1<p<\infty$, and a domain $\Omega\subseteq\mathbb{R}^{n}$. Let $\Omega^{*}\triangleq\Omega\setminus\{\hat{x}\}$ where $\hat{x}\in\Omega$. Under some further conditions, we construct optimal…

Let $\Omega$ be a cone in $\mathbb{R}^{n}$ with $n\ge 2$. For every fixed $\alpha\in\mathbb{R}$ we find the best constant in the Rellich inequality $\int_{\Omega}|x|^{\alpha}|\Delta u|^{2}dx\ge C\int_{\Omega}|x|^{\alpha-4}|u|^{2}dx$ for…

Let $\Omega$ be a domain in $\Ri^d$ with boundary $\Gamma$${\!,}$ $d_\Gamma$ the Euclidean distance to the boundary and $H=-\divv(C\,\nabla)$ an elliptic operator with $C=(\,c_{kl}\,)>0$ where $c_{kl}=c_{lk}$ are real, bounded, Lipschitz…

Let M be an N-function satisfying the $\Delta_2$- condition, let $\omega, \vp$ be two other functions, $\omega\ge 0$. We study Hardy-type inequalities \[ \int_{\rp} M(\omega (x)|u(x)|) {\rm exp}(-\vp (x))dx \le C\int_{\rp} M(|u'(x)|) {\rm…

We prove a simple sufficient criteria to obtain some Hardy inequalities on Riemannian manifolds related to quasilinear second-order differential operator $\Delta_{p}u := \Div(\abs{\nabla u}^{p-2}\nabla u)$. Namely, if $\rho$ is a…

We study semilinear elliptic equations with Hardy potential $\mathrm{(E)} \; -L_\mu u+u^q=0$ in a bounded smooth domain $\Omega\subset \mathbb R^N$. Here $q>1$, $L_\mu=\Delta+\frac{\mu}{\delta_\Omega^2}$ and…

We study the inequality $ -\Delta u - \frac{\mu}{|x|^2} u \geq (|x|^{-\alpha} * u^p)u^q$ in an unbounded cone $\mathcal{C}_\Omega^\rho\subset \mathbb{R}^N$ ($N\geq 2$) generated by a subdomain $\Omega$ of the unit sphere $S^{N-1}\subset…