Search results for "Inequality"

Univariate Grüss- and Ostrowski-Type Inequalities for Positive Linear Operators

2018

Bonnesenʼs inequality for John domains in Rn

2012

Abstract We prove sharp quantitative isoperimetric inequalities for John domains in R n . We show that the Bonnesen-style inequalities hold true in R n under the John domain assumption which rules out cusps. Our main tool is a proof of the isoperimetric inequality for symmetric domains which gives an explicit estimate for the isoperimetric deficit. We use the sharp quantitative inequalities proved in Fusco et al. (2008) [7] and Fuglede (1989) [4] to reduce our problem to symmetric domains.

Nonlinear balayage on metric spaces

2009

We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincaré inequality. In particular, we are interested in continuity and p-harmonicity of the balayage. We also study connections to the obstacle problem. As applications, we characterize regular boundary points and polar sets in terms of balayage. Original Publication:Anders Björn, Jana Björn, Tero Mäkäläinen and Mikko Parviainen, Nonlinear balayage on metric spaces, 2009, Nonlinear Analysis, (71), 5-6, 2153-2171.http://dx.doi.org/10.1016/j.na.2009.01.051Copyright: Elsevier Science B.V., Amsterdam.http://www.elsevier.com/

Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces

2003

Abstract We use the heat equation to establish the Lipschitz continuity of Cheeger-harmonic functions in certain metric spaces. The metric spaces under consideration are those that are endowed with a doubling measure supporting a (1,2)-Poincare inequality and in addition supporting a corresponding Sobolev–Poincare-type inequality for the modification of the measure obtained via the heat kernel. Examples are given to illustrate the necessity of our assumptions on these spaces. We also provide an example to show that in the general setting the best possible regularity for the Cheeger-harmonic functions is Lipschitz continuity.

The De Giorgi measure and an obstacle problem related to minimal surfaces in metric spaces

2010

Abstract We study the existence of a set with minimal perimeter that separates two disjoint sets in a metric measure space equipped with a doubling measure and supporting a Poincare inequality. A measure constructed by De Giorgi is used to state a relaxed problem, whose solution coincides with the solution to the original problem for measure theoretically thick sets. Moreover, we study properties of the De Giorgi measure on metric measure spaces and show that it is comparable to the Hausdorff measure of codimension one. We also explore the relationship between the De Giorgi measure and the variational capacity of order one. The theory of functions of bounded variation on metric spaces is us…

Hitchhiker's guide to the fractional Sobolev spaces

2012

AbstractThis paper deals with the fractional Sobolev spaces Ws,p. We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results.Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.

A quantitative isoperimetric inequality for fractional perimeters

2011

Abstract Recently Frank and Seiringer have shown an isoperimetric inequality for nonlocal perimeter functionals arising from Sobolev seminorms of fractional order. This isoperimetric inequality is improved here in a quantitative form.

Weighted Hardy inequalities beyond Lipschitz domains

2014

It is a well-known fact that in a Lipschitz domain \Omega\subset R^n a p-Hardy inequality, with weight d(x,\partial\Omega)^\beta, holds for all u\in C_0^\infty(\Omega) whenever \beta<p-1. We show that actually the same is true under the sole assumption that the boundary of the domain satisfies a uniform density condition with the exponent \lambda=n-1. Corresponding results also hold for smaller exponents, and, in fact, our methods work in general metric spaces satisfying standard structural assumptions.

Orlicz-Hardy inequalities

2004

We relate Orlicz-Hardy inequalities on a bounded Euclidean domain to certain fatness conditions on the complement. In the case of certain log-scale distortions of Ln, this relationship is necessary and sufficient, thus extending results of Ancona, Lewis, and Wannebo. peerReviewed

Kurdyka–Łojasiewicz–Simon inequality for gradient flows in metric spaces

2019