134,107 research outputs found

    Some inequalities on weighted Sobolev spaces, distance weights and the Assouad dimension

    Full text link
    We study certain inequalities and a related result on weighted Sobolev spaces on bounded John domains in Rn\mathbb{R}^n. Namely, we prove the existence of a right inverse for the divergence operator, along with the corresponding a priori estimate, the improved and the fractional Poincar\'e inequalities, the Korn inequality and the local Fefferman-Stein inequality. All these results are obtained on weighted Sobolev spaces, where the weight is a power of the distance to the boundary. In all cases the exponent of the weight d(,Ω)βpd(\cdot,\partial\Omega)^{\beta p} is only required to satisfy the restriction: βp>(ndimA(Ω))\beta p>-(n-\dim_A(\partial\Omega)), where pp is the exponent of the Sobolev space and dimA(Ω)\dim_A(\partial\Omega) is the Assouad dimension of the boundary of the domain. According to our best knowledge, this condition is less restrictive than the ones in the literature.Comment: 21 pages, 2 figure

    Density problems for second order Sobolev spaces and cut-off functions on manifolds with unbounded geometry

    Get PDF
    We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in W2,pW^{2,p}. The result is improved for p=2p=2 avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calder\'on-Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori-Yau maximum principle for the Hessian.Comment: Improved version. As a main modification, we added a final Section 8 including some additional geometric applications of our result. Furthermore, we proved in Section 7 a disturbed L^p-Sobolev-type inequality with weight more general than the previous one. 25 pages. Comments are welcom

    Is proximity to a food retail store associated with diet and BMI in Glasgow, Scotland?

    Get PDF
    <p><b>Background:</b> Access to healthy food is often seen as a potentially important contributor to diet. Policy documents in many countries suggest that variations in access contribute to inequalities in diet and in health. Some studies, mostly in the USA, have found that proximity to food stores is associated with dietary patterns, body weight and socio-economic differences in diet and obesity, whilst others have found no such relationships. We aim to investigate whether proximity to food retail stores is associated with dietary patterns or Body Mass Index in Glasgow, a large city in the UK.</p> <p><b>Methods:</b> We mapped data from a 'Health and Well-Being Survey' (n = 991), and a list of food stores (n = 741) in Glasgow City, using ArcGIS, and undertook network analysis to find the distance from respondents' home addresses to the nearest fruit and vegetable store, small general store, and supermarket.</p> <p><b>Results:</b> We found few statistically significant associations between proximity to food retail outlets and diet or obesity, for unadjusted or adjusted models, or when stratifying by gender, car ownership or employment.</p> <p><b>Conclusions:</b> The findings suggest that in urban settings in the UK the distribution of retail food stores may not be a major influence on diet and weight, possibly because most urban residents have reasonable access to food stores.</p&gt

    Martin boundary of random walks with unbounded jumps in hyperbolic groups

    Full text link
    Given a probability measure on a finitely generated group, its Martin boundary is a natural way to compactify the group using the Green function of the corresponding random walk. For finitely supported measures in hyperbolic groups, it is known since the work of Ancona and Gou{\"e}zel-Lalley that the Martin boundary coincides with the geometric boundary. The goal of this paper is to weaken the finite support assumption. We first show that, in any non-amenable group, there exist probability measures with exponential tails giving rise to pathological Martin boundaries. Then, for probability measures with superexponential tails in hyperbolic groups, we show that the Martin boundary coincides with the geometric boundary by extending Ancona's inequalities. We also deduce asymptotics of transition probabilities for symmetric measures with superexponential tails

    Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group

    Full text link
    We prove geometric LpL^p versions of Hardy's inequality for the sub-elliptic Laplacian on convex domains Ω\Omega in the Heisenberg group Hn\mathbb{H}^n, where convex is meant in the Euclidean sense. When p=2p=2 and Ω\Omega is the half-space given by ξ,ν>d\langle \xi, \nu\rangle > d this generalizes an inequality previously obtained by Luan and Yang. For such pp and Ω\Omega the inequality is sharp and takes the form \begin{equation} \int_\Omega |\nabla_{\mathbb{H}^n}u|^2 \, d\xi \geq \frac{1}{4}\int_{\Omega} \sum_{i=1}^n\frac{\langle X_i(\xi), \nu\rangle^2+\langle Y_i(\xi), \nu\rangle^2}{\textrm{dist}(\xi, \partial \Omega)^2}|u|^2\, d\xi, \end{equation} where dist(,Ω)\textrm{dist}(\, \cdot\,, \partial \Omega) denotes the Euclidean distance from Ω\partial \Omega.Comment: 14 page

    Lyapunov-type Inequalities for Partial Differential Equations

    Full text link
    In this work we present a Lyapunov inequality for linear and quasilinear elliptic differential operators in NN-dimensional domains Ω\Omega. We also consider singular and degenerate elliptic problems with ApA_p coefficients involving the pp-Laplace operator with zero Dirichlet boundary condition. As an application of the inequalities obtained, we derive lower bounds for the first eigenvalue of the pp-Laplacian, and compare them with the usual ones in the literature
    corecore