arXiv subjects

Ian Fleschler

Publications and source records attributed to Ian Fleschler.

7 recordsLinked to original sources

arXiv2025

An essential one sided boundary singularity for a $3$-dimensional area minimizing current in $\mathbb{R}^5$

We construct a $3$-dimensional area minimizing current $T$ in $\mathbb{R}^5$ whose boundary contains a real analytic surface of multiplicity $2$ at which $T$ has a density $1$ essential boundary singularity with a flat tangent cone. This example shows that the boundary regularity theory we developed with Reinaldo Resende in another paper, which extends Allard's classical boundary regularity result to higher boundary multiplicity, is dimensionally sharp. The construction of $T$ relies on the prescription of boundary data with non-trivial topology, which makes it a flexible technique and gives rise to a wide family of singular examples. In order to understand the examples, we develop a boundary regularity theory for a class of area minimizing $m$-dimensional currents whose boundary consists of smooth $(m-1)$-dimensional surfaces with multiplicities meeting along an $(m-2)$-dimensional smooth submanifold.

math.AP↗

arXiv2024

On the uniqueness of tangent cones to area minimizing currents at boundaries with arbitrary multiplicity

We consider an area minimizing current $T$ in a $C^2$ submanifold $\Sigma$ of $\mathbb{R}^{m+n}$, with arbitrary integer boundary multiplicity $\partial T = Q [\![ \Gamma ]\!]$ where $\Gamma$ is a $C^2$ submanifold of $\Sigma$. We show that at every density $Q/2$ boundary point the tangent cone to $T$ is unique and there is a power rate of convergence to the unique tangent cone. In particular, if $\Gamma$ is a closed manifold which lies at the boundary of a uniformly convex set $\Omega$ and $\Sigma=\mathbb{R}^{m+n}$ then $T$ has a unique tangent cone at every boundary point. As a structural consequence of the uniqueness of the tangent cone, we obtain a decomposition theorem which is the starting point of the boundary regularity theory we develop in another paper in collaboration with Reinaldo Resende. The regularity theory we obtain generalizes Allard's boundary regularity theorem to a higher multiplicity setting.

math.AP↗

arXiv2024

On the regularity of area minimizing currents at boundaries with arbitrary multiplicity

In this paper, we consider an area minimizing integral $m$-current $T$ within a submanifold $\Sigma$ of $\mathbb{R}^{m+n}$, taking a boundary $\Gamma$ with arbitrary multiplicity $Q \in \mathbb{N} \setminus \{0\}$, where $\Gamma$ and $\Sigma$ are $C^{3, \kappa}$. We prove a sharp generalization of Allard's boundary regularity theorem to a higher multiplicity setting. Precisely, we prove that the set of density $Q/2$ singular boundary points of $T$ is $\mathcal{H}^{m-3}$-rectifiable. As a consequence, we show that the entire boundary regular set, without any assumptions on the density, is open and dense in $\Gamma$ which is also dimensionally sharp. Moreover, we prove that if $p \in \Gamma$ admits an open neighborhood in $\Gamma$ consisting of density $Q/2$ points with a tangent cone supported in a half $m$-plane, then $p$ is regular. Furthermore, we show that if the convex barrier condition is satisfied-namely, if $\Gamma$ is a closed manifold that lies at the boundary of a uniformly convex set and $\Sigma = \mathbb{R}^{m+n}$-then the entire boundary singular set is $\mathcal{H}^{m-3}$-rectifiable. Additionally, we investigate certain assumptions on $\Gamma$ that enable us to provide further information about the singular boundary set.

math.AP↗

arXiv2024

Allard-type regularity theory for area minimizing currents at boundaries with arbitrary multiplicity

This is an announcement of a series of upcoming works on boundary regularity for area minimizing currents, one of which is in collaboration with Reinaldo Resende. The setting we consider is that of an area minimizing current with a smooth boundary taken with arbitrary multiplicity. The main result is a generalization of Allard's boundary regularity theorem from which we derive important structural consequences.

math.AP↗

arXiv2023

Carleson's $\varepsilon^2$ conjecture in higher dimensions

In this paper we prove a higher dimensional analogue of Carleson's $\varepsilon^2$ conjecture. Given two arbitrary disjoint open sets $\Omega^+,\Omega^-\subset \mathbb{R}^{n+1}$, and $x\in\mathbb{R}^{n+1}$, $r>0$, we denote $$\varepsilon_n(x,r) := \frac{1}{r^n}\, \inf_{H^+} \mathcal{H}^n \left( ((\partial B(x,r)\cap H^+) \setminus \Omega^+) \cup ((\partial B(x,r)\cap H^-) \setminus \Omega^-)\right),$$ where the infimum is taken over all open affine half-spaces $H^+$ such that $x \in \partial H^+$ and we define $H^-= \mathbb{R}^{n+1} \setminus \overline {H^{+}}$. Our first main result asserts that any Borel subset of $$\left\{x\in\mathbb{R}^{n+1}\, :\, \int_0^1 \varepsilon_n(x,r)^2 \, \frac{dr}{r}<\infty\right\}$$ is $n$-rectifiable. For our second main result we assume that $\Omega^+, \Omega^-$ are open and that $\Omega^+\cup\Omega^-$ satisfies the capacity density condition. For each $x \in \partial \Omega^+ \cup \partial \Omega^-$ and $r>0$, we denote by $\alpha^\pm(x,r)$ the characteristic constant of the (spherical) open sets $\Omega^\pm \cap \partial B(x,r)$. We show that, up to a set of $\mathcal{H}^n$ measure zero, $x$ is a tangent point for both $\partial \Omega^+$ and $ \partial \Omega^-$ if and only if\begin{equation*} \int_0^{1} \min(1,\alpha^+(x,r) + \alpha^-(x,r) -2) \frac{dr}{r} < \infty. \end{equation*} The first result is new even in the plane and the second one improves and extends to higher dimensions the $\varepsilon^2$ conjecture of Carleson.

math.CA↗

arXiv2023

An elementary rectifiability lemma and some applications

We generalize a classical theorem of Besicovitch, showing that, for any positive integers $k<n$, if $E\subset \mathbb R^n$ is a Souslin set which is not $\mathcal{H}^k$-$\sigma$-finite, then $E$ contains a purely unrectifiable closed set $F$ with $0< \mathcal{H}^k (F) < \infty$. Therefore, if $E\subset \mathbb R^n$ is a Souslin set with the property that every closed subset with finite $\mathcal{H}^k$ measure is $k$-rectifiable, then $E$ is $k$-rectifiable. We also point out that this theorem holds in a suitable class of metric spaces. Our interest is motivated by recent studies of the structure of the singular sets of several objects in geometric analysis and we explain the usefulness of our lemma with some examples.

math.CA↗

arXiv2023

Faber-Krahn inequalities, the Alt-Caffarelli-Friedman formula, and Carleson's $\varepsilon^2$ conjecture in higher dimensions

The main aim of this article is to prove quantitative spectral inequalities for the Laplacian with Dirichlet boundary conditions. More specifically, we prove sharp quantitative stability for the Faber-Krahn inequality in terms of Newtonian capacities and Hausdorff contents of positive codimension, thus providing an answer to a question posed by De Philippis and Brasco. One of our results asserts that for any bounded domain $\Omega\subset\mathbb R^n$, $n\geq3$, with Lebesgue measure equal to that of the unit ball $B_0$ and whose first eigenvalue is $\lambda_\Omega$, denoting by $\lambda_{B_0}$ the first eigenvalue for the unit ball, for any $a\in (0,1)$ it holds $$\lambda_\Omega - \lambda_{B_0} \geq C(a) \,\inf_B \bigg(\sup_{t\in (0,1)} \frac1{H^{n-1}(\partial ((1-t) B))} \int_{\partial ((1-t) B)} \frac{\operatorname{Cap}_{n-2}(B(x,atr_B)\setminus \Omega)}{(t\,r_B)^{n-3}}\,dH^{n-1}(x)\bigg)^2,$$ where the infimum is taken over all balls $B$ with the same Lebesgue measure as $\Omega$ and $\operatorname{Cap}_{n-2}$ is the Newtonian capacity of homogeneity $n-2$. In fact, this holds for bounded subdomains of the sphere and the hyperbolic space, as well. In a second result, we also apply the new Faber-Krahn type inequalities to quantify the Hayman-Friedland inequality about the characteristics of disjoint domains in the unit sphere. Thirdly, we propose a natural extension of Carleson's $\varepsilon^2$-conjecture to higher dimensions in terms of a square function involving the characteristics of certain spherical domains, and we prove the necessity of the finiteness of such square function in the tangent points via the Alt-Caffarelli-Friedman monotonicity formula. Finally, we answer in the negative a question posed by Allen, Kriventsov and Neumayer in connection to rectifiability and the positivity set of the ACF monotonicity formula.

math.AP↗