arXiv subjects

Gareth Speight

Publications and source records attributed to Gareth Speight.

29 records · Page 2Linked to original sources

arXiv2017

Domains in metric measure spaces with boundary of positive mean curvature, and the Dirichlet problem for functions of least gradient

We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a $1$-Poincaré inequality. We propose a notion of \emph{domain with boundary of positive mean curvature} and prove that, for such domains, there is always a solution to the Dirichlet problem for least gradients with continuous boundary data. Here \emph{least gradient} is defined as minimizing total variation (in the sense of BV functions) and boundary conditions are satisfied in the sense that the \emph{boundary trace} of the solution exists and agrees with the given boundary data. This extends the result of Sternberg, Williams and Ziemer to the non-smooth setting. Via counterexamples we also show that uniqueness of solutions and existence of \emph{continuous} solutions can fail, even in the weighted Euclidean setting with Lipschitz weights.

math.AP↗

arXiv2016

Porosity, Differentiability and Pansu's Theorem

We use porosity to study differentiability of Lipschitz maps on Carnot groups. Our first result states that directional derivatives of a Lipschitz function act linearly outside a $σ$-porous set. The second result states that irregular points of a Lipschitz function form a $σ$-porous set. We use these observations to give a new proof of Pansu's theorem for Lipschitz maps from a general Carnot group to a Euclidean space.

math.MG↗

arXiv2016

A Measure Zero Universal Differentiability Set in the Heisenberg Group

We show that the Heisenberg group $\mathbb{H}^n$ contains a measure zero set $N$ such that every Lipschitz function $f\colon \mathbb{H}^n \to \mathbb{R}$ is Pansu differentiable at a point of $N$. The proof adapts the construction of small 'universal differentiability sets' in the Euclidean setting: we find a point of $N$ and a horizontal direction where the directional derivative in horizontal directions is almost locally maximal, then deduce Pansu differentiability at such a point.

math.FA↗

arXiv2016

Lusin approximation for horizontal curves in step 2 Carnot groups

A Carnot group $\mathbb{G}$ admits Lusin approximation for horizontal curves if for any absolutely continuous horizontal curve $γ$ in $\mathbb{G}$ and $\varepsilon>0$, there is a $C^1$ horizontal curve $Γ$ such that $Γ=γ$ and $Γ'=γ'$ outside a set of measure at most $\varepsilon$. We verify this property for free Carnot groups of step 2 and show that it is preserved by images of Lie group homomorphisms preserving the horizontal layer. Consequently, all step 2 Carnot groups admit Lusin approximation for horizontal curves.

math.MG↗

arXiv2015

Lusin Approximation and Horizontal Curves in Carnot Groups

We show that, given an absolutely continuous horizontal curve $γ$ in the Heisenberg group, there is a $C^1$ horizontal curve $Γ$ such that $Γ=γ$ and $Γ'=γ'$ outside a set of small measure. Conversely, we construct an absolutely continuous horizontal curve in the Engel group with no $C^1$ horizontal approximation.

math.MG↗

arXiv2014

The p-Weak Gradient Depends on p

Given a>0, we construct a weighted Lebesgue measure on R^n for which the family of non constant curves has p-modulus zero for p\leq 1+a but the weight is a Muckenhoupt A_p weight for p>1+a. In particular, the p-weak gradient is trivial for small p but non trivial for large p. This answers an open question posed by several authors. We also give a full description of the p-weak gradient for any locally finite Borel measure on the real line.

math.FA↗

arXiv2014

Tensorization of Cheeger energies, the space $H^{1,1}$ and the area formula for graphs

First we study in detail the tensorization properties of weak gradients in metric measure spaces $(X,d,m)$. Then, we compare potentially different notions of Sobolev space $H^{1,1}(X,d,m)$ and of weak gradient with exponent 1. Eventually we apply these results to compare the area functional $\int\sqrt{1+|\nabla f|_w^2}\,dm$ with the perimeter of the subgraph of $f$, in the same spirit as the classical theory.

math.FA↗

arXiv2014

Differentiability of Lipschitz Functions in Lebesgue Null Sets

We show that if n>1 then there exists a Lebesgue null set in R^n containing a point of differentiability of each Lipschitz function mapping from R^n to R^(n-1); in combination with the work of others, this completes the investigation of when the classical Rademacher theorem admits a converse. Avoidance of sigma-porous sets, arising as irregular points of Lipschitz functions, plays a key role in the proof.

math.FA↗

arXiv2012

Surfaces Meeting Porous Sets in Positive Measure

Let n>2 and X be a Banach space of dimension strictly greater than n. We show there exists a directionally porous set P in X for which the set of C^1 surfaces of dimension n meeting P in positive measure is not meager. If X is separable this leads to a decomposition of X into a countable union of directionally porous sets and a set which is null on residually many C^1 surfaces of dimension n. This is of interest in the study of certain classes of null sets used to investigate differentiability of Lipschitz functions on Banach spaces.

math.FA↗

arXiv2012

Directional Lower Porosity

We investigate differences between upper and lower porosity. In finite dimensional Banach spaces every upper porous set is directionally upper porous. We show the situation is very different for lower porous sets; there exists a lower porous set in the plane which is not even a countable union of directionally lower porous sets.

math.FA↗

arXiv2011

Differentiability, Porosity and Doubling in Metric Measure Spaces

We show if a metric measure space admits a differentiable structure then porous sets have measure zero and hence the measure is pointwise doubling. We then give a construction to show if we only require an approximate differentiable structure the measure need no longer be pointwise doubling.

math.MG↗