arXiv subjects

Giacomo Del Nin

Publications and source records attributed to Giacomo Del Nin.

20 records · Page 2Linked to original sources

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↗

arXiv2018

Sticky-disk limit of planar $N$-bubbles

We study planar $N$-clusters that minimize, under an area constraint, a weighted perimeter $P_\varepsilon$ depending on a small parameter $\varepsilon>0$. Specifically we weight $2-\varepsilon$ the boundary between the interior chambers and $1$ the boundary between an interior chamber and the exterior one. We prove that as $\varepsilon\to 0$ minimizers of $P_\varepsilon$ converge to configurations of disjoint disks that maximize the number of tangencies, each weighted by the harmonic mean of the radii of the two tangent disks. We also obtain some information on the structure of minimizers for small $\varepsilon$.

math.OC↗