arXiv subjects

Kennedy Obinna Idu

Publications and source records attributed to Kennedy Obinna Idu.

6 recordsLinked to original sources

arXiv2026

On rectifiability in Heisenberg groups without a lower density condition

We resolve a problem posed by Mattila, Serapioni and Serra Cassano concerning the role of density assumptions in the characterization of rectifiable sets of low codimension in Heisenberg groups. Specifically, we prove that the positive lower density condition is not required: rectifiability is completely determined by the geometric property of almost everywhere existence of approximate tangent subgroups. This provides a simplified and intrinsic criterion for rectifiable sets in the sub-Riemannian setting and sharpens the analogy with Federer's classical Euclidean theorem.

math.MG↗

arXiv2024

The planar Plateau's problem via capillarity

The Plateau's problem seeks to determine a surface of minimal area which spans a given boundary. It is widely studied for its varied mathematical formulations, applications and relevance to physical models such as soap films. We revisit the problem and study a soap film model in the spirit of capillarity formulations in two dimensions. Our approach introduces a nonlocal geometric potential in the variational length minimization scheme. This incorporates effects of thickness of soap films and provides insight into addressing the so-called collapsing phenomenon and other observable physical phenomena and properties.

math.CA↗

arXiv2024

Alexandrov's estimate revisited

Alexandrov's estimate states that if $Ω$ is a bounded open convex domain in ${\mathbb R}^n$ and $u:\bar Ω\to {\mathbb R}$ is a convex solution of the Monge-Ampere equation $\det D^2 u = f$ that vanishes on $\partial Ω$, then \[ |u(x) - u(y)| \le ω(|x-y|)(\int_Ωf)^{1/n} \qquad \mbox{for }ω(δ) = C_n\,\mbox{diam}(Ω)^{\frac{n-1}n} δ^{1/n}. \] We establish a variety of improvements of this, depending on the geometry of $\partial Ω$. For example, we show that if the curvature is bounded away from $0$, then the estimate remains valid if $ω(δ)$ is replaced by $C_Ωδ^{\frac 12 + \frac 1{2n}}$. We determine the sharp constant $C_Ω$ when $n=2$, and when $n\ge 3$ and $\partial Ω$ is $C^2$, we determine the sharp asymptotics of the optimal modulus of continuity $ω_Ω(δ)$ as $δ\to 0$. For arbitrary convex domains, we characterize the scaling of the optimal modulus $ω_Ω$. Under very mild nondegeneracy conditions, our results yield the improved Holder estimate, $ω_Ω(δ) \le C δ^α$ for some $α>1/n$.

math.AP↗

arXiv2023

$C^{1,α}$-rectifiability in low codimension in Heisenberg groups

A natural notion of higher order rectifiability is introduced for subsets of Heisenberg groups $\mathbb{H}^n$ in terms of covering a set almost everywhere by a countable union of $(\mathbf{C}_H^{1,α},\mathbb{H})$-regular surfaces, for some $0 < α\leq 1$. We prove that a sufficient condition for $C^{1,α}$-rectifiability of low-codimensional subsets in Heisenberg groups is the almost everywhere existence of suitable approximate tangent paraboloids.

math.MG↗

arXiv2020

Characterizations of $k$-rectifiability in homogenous groups

A well known notion of $k$-rectifiable set can be formulated in any metric space using Lipschitz images of subsets of $\mathbb{R}^k$. We prove some characterizations of $k$-rectifiability, when the metric space is an arbitrary homogeneous group. In particular, we show that the a.e. existence of the $(k,\mathbb{G})$-approximate tangent group implies $k$-rectifiability.

math.MG↗

arXiv2019

Geometric criteria for $C^{1,α}$ rectifiability

We prove criteria for $\mathcal{H}^k$-rectifiability of subsets of $\mathbb{R}^n$ with $C^{1,α}$ maps, $0<α\leq 1$, in terms of suitable approximate tangent paraboloids. We also provide a version for the case when there is not an a priori tangent plane, measuring on dyadic scales how close the set is to lying in a $k$-plane. We then discuss the relation with similar criteria involving Peter Jones' $β$ numbers, in particular proving that a sufficient condition is the boundedness for small $r$ of $r^{-α}β_p(x,r)$ for $\mathcal{H}^k$-a.e. $x$ and for any $1\leq p\leq \infty$.

math.CA↗