On a control problem for two orthogonal nets of curves in the plane

Guy David; Polina Perstneva

Home/Archive/arXiv

arXiv · 2610.11569

On a control problem for two orthogonal nets of curves in the plane

Guy David·Polina Perstneva

View original source ↗Read source document ↗Export metadata ↓

Abstract

We construct nets of curves in a rectangle $R$ with special properties. The net is formed by two families of curves, "horizontal" and "vertical", each family foliates $R$, and every horizontal curve meets every vertical curve orthogonally. Moreover, every horizontal curve started at the left edge of $R$ arrives to the right edge of $R$ at the same height, and every vertical curve started at the bottom edge of $R$ arrives to the top edge of $R$ at a prescribed in advance longitude. The latter longitude function as a map from bottom to top can be chosen in advance as any small perturbation of the identity. Our motivation for this was to design some elliptic divergence form operators, for instance on one of the domains bounded by a Reifenberg flat curve in the plane, with the scalar coefficient and the corresponding elliptic measure being absolutely continuous with respect to the natural measure on the boundary (an object of a separate paper). The construction of our net of curves in a rectangle can also be interpreted as control problem, has an interest on its own, and, likely, has other applications.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Guy David, Polina Perstneva. 2026-10-08. On a control problem for two orthogonal nets of curves in the plane. https://arxiv.org/abs/2610.11569

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Selector form of Weaver's conjecture, Feichtinger's conjecture, and frame sparsification

We show an extension of a probabilistic result of Marcus, Spielman, and Srivastava, which resolved the Kadison-Singer problem, for block diagonal positive semidefinite random matrices. We use this result to show several selector results, which generalize their partition counterparts. This includes a selector form of Weaver's KS$_r$ conjecture for block diagonal trace class operators, which extends a selector result for Bessel sequences, or equivalently rank one matrices, due to Londner and the author. We also show a selector variant of Feichtinger's conjecture for a (possibly infinite) collection of Bessel sequences, extending earlier results for a single Bessel sequence. We prove a generalization of the $R_ε$ conjecture of Casazza, Tremain, and Vershynin for infinite collection of equal norm Bessel sequences. In particular, our selector result yields a conjectured asymptotically optimal bound for a single Bessel sequence in terms of Riesz sequence tightness parameter. We establish an iterated selector form of Weaver's KS$_2$ conjecture and show its applications. This includes a solution of an open problem on nearly unit norm Parseval frames of exponentials, which was posed by Londner and the author. We generalize a discretization result for continuous frames by Freeman and Speegle in two ways. First, we extend their result from the setting of rank one operators to positive trace operator valued measures. Second, we establish a nearly tight discretization of bounded continuous Parseval frames. In particular, our selector result yields an improvement of the result of Nitzan, Olevskii, and Ulanovskii and implies the existence of nearly tight exponential frames for unbounded sets with an explicit control on their frame redundancy.

math.CA↗

arXiv2026

Hausdorff-Choquet Angular Spaces and Weak-Type $(1,1)$ Bounds for Rough Maximal Operators

In the present paper, we consider the maximal operator \[ \mathcal M_Ωf(x) := \sup_{r>0}\frac{1}{r^n} \int_{|y| 2$. To our knowledge, the present paper is the first to relax this threshold for $n>2$. Finally, for the general Hausdorff--Choquet scale $\mathcal H\mathcal C_α$, we show that $α=(n-1)/2$ is the sharp exponent for uniform weak type $(1,1)$ estimates.

math.CA↗

arXiv2026

Revisiting Positive Exponent power weighted Stein-Weiss inequalities in $L^1$

We obtain the necessary and sufficient conditions on input functions for a family of positive exponent power weight $L^1$ Stein-Weiss inequalities. Earlier work by De Nápoli and Picon showed that a certain cocanceling condition was necessary and sufficient in $L^1$, but it was unknown if this condition remained necessary in the positive exponent power weight setting. This present article identifies weaker necessary conditions for positive exponent power weights, some of which were shown to be sufficient for positive, non-integer exponent power weights in the author's earlier work.

math.CA↗