Search results for "Equality."
showing 10 items of 1308 documents
Exceptional Sets for Quasiconformal Mappings in General Metric Spaces
2008
A theorem of Balogh, Koskela, and Rogovin states that in Ahlfors Q-regular metric spaces which support a p-Poincare inequality, , an exceptional set of -finite (Q−p)- dimensional Hausdorff measure can be taken in the definition of a quasiconformal mapping while retaining Sobolev regularity analogous to that of the Euclidean setting. Through examples, we show that the assumption of a Poincare inequality cannot be removed.
A maximal Function Approach to Two-Measure Poincaré Inequalities
2018
This paper extends the self-improvement result of Keith and Zhong in Keith and Zhong (Ann. Math. 167(2):575–599, 2008) to the two-measure case. Our main result shows that a two-measure (p, p)-Poincare inequality for $$10$$ under a balance condition on the measures. The corresponding result for a maximal Poincare inequality is also considered. In this case the left-hand side in the Poincare inequality is replaced with an integral of a sharp maximal function and the results hold without a balance condition. Moreover, validity of maximal Poincare inequalities is used to characterize the self-improvement of two-measure Poincare inequalities. Examples are constructed to illustrate the role of t…
Cheeger-harmonic functions in metric measure spaces revisited
2014
Abstract Let ( X , d , μ ) be a complete metric measure space, with μ a locally doubling measure, that supports a local weak L 2 -Poincare inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic functions are Lipschitz continuous on ( X , d , μ ) . Gradient estimates for Cheeger-harmonic functions and solutions to a class of non-linear Poisson type equations are presented.
An isoperimetric type problem for primitive Pythagorean hodograph curves
2012
An isoperimetric type problem for primitive Pythagorean hodograph curves is studied. We show how to compute, for each possible degree, the Pythagorean hodograph curve of a given perimeter enclosing the greatest area. We also discuss the existence and construction of smooth solutions, obtaining a relationship with an interesting sequence of Appell polynomials.
Neumann p-Laplacian problems with a reaction term on metric spaces
2020
We use a variational approach to study existence and regularity of solutions for a Neumann p-Laplacian problem with a reaction term on metric spaces equipped with a doubling measure and supporting a Poincare inequality. Trace theorems for functions with bounded variation are applied in the definition of the variational functional and minimizers are shown to satisfy De Giorgi type conditions.
Sobolev Spaces and Quasiconformal Mappings on Metric Spaces
2001
Heinonen and I have recently established a theory of quasiconformal mappings on Ahlfors regular Loewner spaces. These spaces are metric spaces that have sufficiently many rectifiable curves in a sense of good estimates on moduli of curve families. The Loewner condition can be conveniently described in terms of Poincare inequalities for pairs of functions and upper gradients. Here an upper gradient plays the role that the length of the gradient of a smooth function has in the Euclidean setting. For example, the Euclidean spaces and Heisenberg groups and the more general Carnot groups admit the type of a Poincare inequality we need. We describe the basics and discuss the associated Sobolev sp…
Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems
2013
In this paper we prove a sharp lower bound for the first non-trivial Neumann eigenvalue μ1(Ω) for the p-Laplace operator (p > 1) in a Lipschitz bounded domain Ω in ℝn. Our estimate does not require any convexity assumption on Ω and it involves the best isoperimetric constant relative to Ω. In a suitable class of convex planar domains, our bound turns out to be better than the one provided by the Payne—Weinberger inequality.
Admissibility versus Ap-Conditions on Regular Trees
2020
We show that the combination of doubling and (1, p)-Poincaré inequality is equivalent to a version of the Ap-condition on rooted K-ary trees. peerReviewed
Qualidade docente e eficácia escolar
2008
Este texto visa a explorar uma faceta do funcionamento do sistema escolar que permite associar a implementação de políticas públicas, no caso, a distribuição dos professores pelas escolas e salas de aula, e à avaliação dos impactos dessa distribuição do ponto de vista da busca por maior igualdade de oportunidades. Para discutir essa questão, usaremos os dados de pesquisa realizada no Brasil, na Argentina, no Chile e no México, com informações de 96 escolas e professores, assim como de 2048 alunos de 4ª série que estudavam em escolas situadas em bairros pobres de Belo Horizonte, Buenos Aires, Santiago e Guanajuato. Inicialmente, discutimos alguns dos elementos das políticas educacionais lati…
Nonequilibrium critical scaling in quantum thermodynamics
2016
The emerging field of quantum thermodynamics is contributing important results and insights into archetypal many-body problems, including quantum phase transitions. Still, the question whether out-of-equilibrium quantities, such as fluctuations of work, exhibit critical scaling after a sudden quench in a closed system has remained elusive. Here, we take a novel approach to the problem by studying a quench across an impurity quantum critical point. By performing density matrix renormalization group computations on the two-impurity Kondo model, we are able to establish that the irreversible work produced in a quench exhibits finite-size scaling at quantum criticality. This scaling faithfully …