134,107 research outputs found
Some inequalities on weighted Sobolev spaces, distance weights and the Assouad dimension
We study certain inequalities and a related result on weighted Sobolev spaces
on bounded John domains in . 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 is only required to satisfy the
restriction: , where is the exponent
of the Sobolev space and 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
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 . The
result is improved for 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?
<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>
Martin boundary of random walks with unbounded jumps in hyperbolic groups
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
We prove geometric versions of Hardy's inequality for the sub-elliptic
Laplacian on convex domains in the Heisenberg group ,
where convex is meant in the Euclidean sense. When and is the
half-space given by this generalizes an
inequality previously obtained by Luan and Yang. For such and 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 denotes the
Euclidean distance from .Comment: 14 page
Lyapunov-type Inequalities for Partial Differential Equations
In this work we present a Lyapunov inequality for linear and quasilinear
elliptic differential operators in dimensional domains . We also
consider singular and degenerate elliptic problems with coefficients
involving the Laplace operator with zero Dirichlet boundary condition.
As an application of the inequalities obtained, we derive lower bounds for
the first eigenvalue of the Laplacian, and compare them with the usual ones
in the literature
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