Related papers: Poincare Inequalities in Punctured Domains

In this paper we prove that if $\Omega\in\mathbb{R}^n$ is a bounded John domain, the following weighted Poincare-type inequality holds: $$ \inf_{a\in \mathbb{R}}\| (f(x)-a) w_1(x) \|_{L^q(\Omega)} \le C \|\nabla f(x) d(x)^\alpha w_2(x)…

The classical Poincar\'e inequality establishes that for any bounded regular domain $\Omega\subset \R^N$ there exists a constant $C=C(\Omega)>0$ such that $$ \int_{\Omega} |u|^2\, dx \leq C \int_{\Omega} |\nabla u|^2\, dx \ \ \forall u \in…

We first define the trace on a domain $\Omega$ which is definable in an o-minimal structure. We then show that every function $u\in W^{1,p}(\Omega)$ vanishing on the boundary in the trace sense satisfies Poincar\'e inequality. We finally…

We study Poincar\'e-Wirtinger type inequalities in the framework of magnetic fractional Sobolev spaces. In the local case, Lieb-Seiringer-Yngvason [E. Lieb, R. Seiringer, and J. Yngvason, Poincar\'e inequalities in punctured domains, Ann.…

Let $\Omega \subset \mathbb{R}^n$ be a convex. If $u: \Omega \rightarrow \mathbb{R}$ has mean 0, then we have the classical Poincar\'{e} inequality $$ \|u \|_{L^p} \leq c_p \mbox{diam}(\Omega) \| \nabla u \|_{L^p}$$ with sharp constants…

We prove a fractional version of Poincar\'e inequalities in the context of $\R^n$ endowed with a fairly general measure. Namely we prove a control of an $L^2$ norm by a non local quantity, which plays the role of the gradient in the…

The classical Poincar\'e estimate establishes closedness of the range of the gradient in unweighted $L^2(\Omega)$-spaces as long as $\Omega\subseteq\mathbb{R}^3$ is contained in a slab, that is, $\Omega$ is bounded in one direction. Here,…

We study first order differential operators with constant coefficients. The main question is under what conditions a generalized Poincar\'e inequality holds. We show that the constant rank condition is sufficient. The concept of the…

In the paper, we investigate Poincare type inequalities for the functions having zero mean value on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. We derive exact and easily computable constants for some…

In this paper we investigate the fractional Poincar\'e inequality on unbounded domains. In the local case, Sandeep-Mancini showed that in the class of simply connected domains, Poincar\'e inequality holds if and only if the domain does not…

We prove Sobolev-Poincar\'e and Poincar\'e inequalities in variable Lebesgue spaces $L^{p(\cdot)}(\Omega)$, with $\Omega\subset{\mathbb R}^n$ a bounded John domain, with weaker regularity assumptions on the exponent $p(\cdot)$ that have…

For any convex set $\Omega \subset {\mathbb R} ^N$, we provide a lower bound for the inverse of the Poincar\'e constant in $W ^ {1, 1}(\Omega)$: it refines an inequality in terms of the diameter due to Acosta-Duran, via the addition of an…

If $\Omega \subset \R^n$ is a smooth bounded domain and $q \in (0, \frac{n}{n-1})$ we consider the Poincare-Sobolev inequality \[ c \Bigl(\int_{\Omega} \abs{u}^\frac{n}{n-1}\Bigr)^{1-\frac{1}{n}} \le \int_{\Omega} \abs{Du}, \] for every $u…

In the context of Sobolev spaces with variable exponents, Poincar\'e--Wirtinger inequalities are possible as soon as Luxemburg norms are considered. On the other hand, modular versions of the inequalities in the expected form…

Let $\Omega \subset \rr^2$ be a bounded simply connected domain. We show that, for a fixed (every) $p\in (1,\fz),$ the divergence equation $\mathrm{div}\,\mathbf{v}=f$ is solvable in $W^{1,p}_0(\Omega)^2$ for every $f\in L^p_0(\Omega)$, if…

The Poincare Inequality is an extremely useful tool in the analysis of PDEs. A significant amount of literature has dealt with finding the optimal constant $C(p,\Omega)$, depending only the domain $\Omega$, and the $L^p$ norm in question.…

Given a bounded open subset $\Omega$ of $\mathbb R^n$, we establish the weak closure of the affine ball $B^{\mathcal A}_p(\Omega) = \{f \in W^{1,p}_0(\Omega):\ \mathcal E_p f \leq 1\}$ with respect to the affine functional $\mathcal E_pf$…

In this article, we study the generalized Poincare problem from the opposite perspective, by establishing lower bounds on the degree of the vector field in terms of invariants of the variety.

The paper is concerned with sharp estimates of constants in Poincare type inequalities for functions having zero mean value on the boundary of a Lipschitz domain or on a measurable part of it. These estimates are useful for various…

We study first order differential operators with constant coefficients. The main question is under what conditions a generalized Poincar\'e inequality holds. We show that the constant rank condition is sufficient. The concept of the…