Related papers: Poincare Inequalities in Punctured Domains

Let $\Omega$ be a bounded domain of $\mathbb{R}^{n+1}$ with $n \ge 1$. We assume that the boundary $\Gamma$ of $\Omega$ is Lipschitz. Consider the Dirichlet-to-Neumann operator $N_0$ associated with a system in divergence form of size $m$…

We prove a sharp upper bound on convex domains, in terms of the diameter alone, of the best constant in a class of weighted Poincar\'e inequalities. The key point is the study of an optimal weighted Wirtinger inequality.

For a smooth bounded domain $\Omega$ and $p \geq q \geq 2$, we establish quantified versions of the classical Friedrichs inequality $\|\nabla u\|_p^p - \lambda_1 \|u\|_q^p \geq 0$, $u \in W_0^{1,p}(\Omega)$, where $\lambda_1$ is a…

In this paper we consider the functional \begin{equation*} E_{p,\la}(\Omega):=\int_\Omega \dist^p(x,\pd \Omega )\d x+\la \frac{\H^1(\pd \Omega)}{\H^2(\Omega)}. \end{equation*} Here $p\geq 1$, $\la>0$ are given parameters, the unknown…

Given a smooth positive function $f$ defined on the unit circle satisfying a simple condition, we obtain a Poincar\'{e}-type inequality for an arbitrary function $u$ whose weighted average with respect to $f$ is zero. The proof uses…

We prove generalizations of the Poincare and logarithmic Sobolev inequalities corresponding to the case of fractional derivatives in measure spaces with only a minimal amount of geometric structure. The class of such spaces includes (but is…

We study topological Poincar\'e type inequalities on general graphs. We characterize graphs satisfying such inequalities and then turn to the best constants in these inequalities. Invoking suitable metrics we can interpret these constants…

In this short note we consider several widely used L^2-orthogonal Helmholtz decompositions for bounded domains in R^3. It is well known that one part of the decompositions is a subspace of the space of functions with zero mean. We refine…

The celebrated Poincar\'e and Friedrichs inequalities estimate the $\mathbb{L}_p$-norm of a function by the $\mathbb{L}_p$-norm of the gradient. We prove the Poincar\'e inequality for a domain $\Omega\subset \mathbb{R}^n$ and for a…

We study Poincar\'e type $L^p$ inequality on a compact semialgebraic subset of $\R^n$ for $p>>1$. First we derive a local inequality by using a Lipschitz deformation retraction with estimates on its derivatives. Then, we extend the local…

We make explicit the $p$-dependence of $C$ in the gradient estimate $\left\Vert \nabla u\right\Vert _{\infty}^{p-1}\leq C\left\Vert f\right\Vert _{N,1}$ by Cianchi and Maz'ya (2011). In such inequality, the constant $C$ is uniform with…

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…

We study the dependence of the continuity constants for the regularized Poincar\'e and Bogovski\u{\i} integral operators acting on differential forms defined on a domain $\Omega$ of $\mathbb{R}^n$. We, in particular, study the dependence of…

We study an inhomogeneous Neumann boundary value problem for functions of least gradient on bounded domains in metric spaces that are equipped with a doubling measure and support a Poincar\'e inequality. We show that solutions exist under…

A conjecture of Pasteczka, generalizing the classical Hermite--Hadamard Inequality, states that if $\Omega \subseteq \mathbb{R}^d$ is a compact convex domain such that $\Omega$ and $\partial \Omega$ have the same center of mass, then for…

In this note we show that gradient of Harmonic functions on a smooth domain with Lipschitz boundary values is pointwise bounded by a universal function which is in $L^p$ for all finite $p\geq 1$.

Given a bounded domain $\Omega \subset {\mathbb R}^{n}$ with $n\ge2$, let $\phi $ is a Young function satisfying the doubling condition with the constant $K_\phi<2^{n}$. If $\Omega$ is a John domain, we show that $\Omega $ supports a…

In this article, we deal with the fine boundary regularity, a weighted H\"{o}lder regularity of weak solutions to the problem involving the fractional $(p,q)$ Laplacian denoted by $(-\Delta)_{p}^{s} u + (-\Delta)_{q}^{s} u = f(x)$ in…

We consider a \emph{family} $(P_\omega)_{\omega \in \Omega}$ of elliptic second order differential operators on a domain $U_0 \subset \mathbb{R}^m$ whose coefficients depend on the space variable $x \in U_0$ and on $\omega \in \Omega,$ a…

Within the setting of metric spaces equipped with a doubling measure and supporting a $p$-Poincar\'e inequality, establishing existence of solutions to Dirichlet problem in a bounded domain in such a metric space is accomplished via direct…