Related papers: Rellich inequalities with weights

This paper is devoted to the Moser-Trudinger-Onofri inequality on smooth compact connected Riemannian manifolds. We establish a rigidity result for the Euler-Lagrange equation and deduce an estimate of the optimal constant in the inequality…

We prove the sharp inequality \[ J(\Omega) := \frac{\lambda_1(\Omega)}{h_1(\Omega)^2} < \frac{\pi^2}{4},\] where $\Omega$ is any planar, convex set, $\lambda_1(\Omega)$ is the first eigenvalue of the Laplacian under Dirichlet boundary…

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…

Let $\Delta_0$ be the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$. We show that the Hardy-Rellich inequality of the form $$ \int_{\mathbb{S}^{d-1}} \left | f (x)\right|^2 d\sigma(x) \leq c_d \min_{e\in…

Let $\Omega$ be a bounded $C^{2}$ domain in $\R^n$, and let $\Omega^{\ast}$ be the Euclidean ball centered at 0 and having the same Lebesgue measure as $\Omega$. Consider the operator $L=-\div(A\nabla)+v\cdot \nabla +V$ on $\Omega$ with…

In this paper we obtain the best constants in some higher order Sobolev inequalities in the critical exponent. These inequalities can be separated into two types: those that embed into $L^\infty(\mathbb{R}^N)$ and those that embed into…

In this paper, we study Hardy's inequality in a limiting case: $$ \int_{\Omega} |\nabla u |^N dx \ge C_N(\Omega) \int_{\Omega} \frac{|u(x)|^N}{|x|^N \left(\log \frac{R}{|x|} \right)^N} dx $$ for functions $u \in W^{1,N}_0(\Omega)$, where…

We show several variants of concentration inequalities on the sphere stated as subgaussian estimates with optimal constants. For a Lipschitz function, we give one-sided and two-sided bounds for deviation from the median as well as from the…

Let $\Omega\subset \mathbb R^{n+1}$, $n\geq1$, be a bounded open set satisfying the interior corkscrew condition with a uniformly $n$-rectifiable boundary but without any connectivity assumptions. We establish the estimate $$ \Vert…

We prove a Payne-Weinberger type inequality for the $p$-Laplacian Neumann eigenvalues ($p\ge 2$). The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of the best constants in Poincar\'e…

Let $w_\alpha(t) := t^{\alpha}\,e^{-t}$, where $\alpha > -1$, be the Laguerre weight function, and let $\|\cdot\|_{w_\alpha}$ be the associated $L_2$-norm, $$ \|f\|_{w_\alpha} = \left\{\int_{0}^{\infty} |f(x)|^2…

Let $\Omega$ be an open subset of $\mathbb{R}^N$ with $N\geq 2.$ We identify various classes of Young functions $\Phi$ and $\Psi$, and function spaces for a weight function $g$ so that the following weighted Orlicz-Sobolev inequality holds:…

Let $\Omega$ be an unbounded open subset of ${\mathbb R}^n$, $n \ge 2$, and $A : \Omega \times {\mathbb R}^n \to {\mathbb R}^n$ be a function such that $$ C_1 |\zeta|^p \le \zeta A (x, \zeta), \quad |A (x, \zeta)| \le C_2 |\zeta|^{p-1} $$…

We prove two-sided estimates for the best (i.e., the smallest possible) constant $\,c_n(\alpha)\,$ in the Markov inequality $$ \|p_n'\|_{w_\alpha} \le c_n(\alpha) \|p_n\|_{w_\alpha}\,, \qquad p_n \in {\cal P}_n\,. $$ Here, ${\cal P}_n$…

Let $S^2$ be the 2-dimensional unit sphere and let $J_\alpha $ denote the nonlinear functional on the Sobolev space $H^{1,2}(S^2)$ defined by $$ J_\alpha(u) = \frac{\alpha}{4}\int_{S^2}|\nabla u|^2 d\omega + \int_{S^2} u d\omega -\ln…

We study Rellich inequalities associated to higher-order elliptic operators in the Euclidean space. The inequalities are expressed in terms of an associated Finsler metric. In the case of half-spaces we obtain the sharp constant while for a…

We study a minimizing problem associated with the singular problem \[ \left\{ \begin{array} [c]{ll} -\operatorname{div}\left( \left\vert \nabla u\right\vert ^{p-2}\nabla u\right) =\lambda u^{-1} & \mathrm{in\ }\Omega\\ u>0 & \mathrm{in\…

We consider second-order uniformly elliptic operators subject to Dirichlet boundary conditions. Such operators are considered on a bounded domain $\Omega$ and on the domain $\phi(\Omega)$ resulting from $\Omega$ by means of a bi-Lipschitz…

Refining the sharp upper bounds $\mu_{k,d}^* $ obtained by Kr\"oger (1999) for the $k$-th Neumann eigenvalue of a convex domain $\Omega \subset \mathbb{R}^d$, we prove the following inequalities: for any $k\in \mathbb{N}$ there exists a…

We investigate the nonnegative solutions to the nonlinear integral inequality $u \ge I_{\alpha}\ast\big((I_\beta \ast u^p)u^q\big)$ a.e. in $\mathbb{R}^N$, where $\alpha, \beta\in (0,N)$, $p, q>0$ and $I_\alpha$, $I_\beta$ denote the Riesz…