Search results for "30c65"

Harmonic morphisms in nonlinear potential theory

1992

This article concerns the following problem: given a family of partial differential operators with similar structure and given a continuous mapping f from an open set Ω in Rn into Rn, then when does f pull back the solutions of one equation in the family to solutions of another equation in that family? This problem is typical in the theory of differential equations when one wants to use a coordinate change to study solutions in a different environment.

Duality of moduli in regular toroidal metric spaces

2020

We generalize a result of Freedman and He [4, Theorem 2.5], concerning the duality of moduli and capacities in solid tori, to sufficiently regular metric spaces. This is a continuation of the work of the author and Rajala [12] on the corresponding duality in condensers. peerReviewed

Sharp capacity estimates for annuli in weighted R^n and in metric spaces

2017

We obtain estimates for the nonlinear variational capacity of annuli in weighted R^n and in metric spaces. We introduce four different (pointwise) exponent sets, show that they all play fundamental roles for capacity estimates, and also demonstrate that whether an end point of an exponent set is attained or not is important. As a consequence of our estimates we obtain, for instance, criteria for points to have zero (resp. positive) capacity. Our discussion holds in rather general metric spaces, including Carnot groups and many manifolds, but it is just as relevant on weighted R^n. Indeed, to illustrate the sharpness of our estimates, we give several examples of radially weighted R^n, which …

Differentiability in the Sobolev space W1,n-1

2014

Let Ω ⊂ Rn be a domain, n ≥ 2. We show that a continuous, open and discrete mapping f ∈ W1,n−1 loc (Ω, Rn ) with integrable inner distortion is differentiable almost everywhere on Ω. As a corollary we get that the branch set of such a mapping has measure zero. peerReviewed

Uniformization with infinitesimally metric measures

2019

We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces $X$ homeomorphic to $\mathbb R^2$. Given a measure $\mu$ on such a space, we introduce $\mu$-quasiconformal maps $f:X \to \mathbb R^2$, whose definition involves deforming lengths of curves by $\mu$. We show that if $\mu$ is an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a $\mu$-quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinitesimally quasisymmetric parametrization.

Generalized John disks

2014

Abstract We establish the basic properties of the class of generalized simply connected John domains.

A Newman property for BLD-mappings

2019

We define a Newman property for BLD-mappings and prove that for a BLD-mapping between generalized manifolds equipped with complete path-metrics, this property is equivalent to the branch set being porous when the codomain is LLC. peerReviewed

Sharpness of Rickman’s Picard theorem in all dimensions

2015

We show that given \({n \geqslant 3}\), \({q \geqslant 1}\), and a finite set \({\{y_1, \ldots, y_q \}}\) in \({\mathbb{R}^n}\) there exists a quasiregular mapping \({\mathbb{R}^n\to \mathbb{R}^n}\) omitting exactly points \({y_1, \ldots, y_q}\).

Mappings of finite distortion: Sharp Orlicz-conditions

2003

We establish continuity, openness and discreteness, and the condition $(N)$ for mappings of finite distortion under minimal integrability assumptions on the distortion.

A note on mappings of finite distortion: The sharp modulus of continuity

2005