Search arXiv⌕ Search

arXiv subjects

Max Goering

Publications and source records attributed to Max Goering.

15 recordsLinked to original sources
arXiv2025

On the geometric properties of multi-operator two-phase elliptic measure

We provide a structural characterization of a given boundary using two-phase elliptic measure in a multi-operator setting, extending to this novel setting results of Kenig, Preiss & Toro, Toro & Zhao and Azzam & Mourgoglou, including the validity of Oksendal's conjecture for two-sided NTA domains under the assumption of mutual absolute continuity of the elliptic measures. Our techniques rely on a reduction to a multi-operator two-phase free-boundary problem combined with an extension of the powerful tools introduced by Preiss in his Density Theorem.

math.AP↗
arXiv2024

On the dimension of $s$-Nikod\'ym sets

Let $s \in [0,1]$. We show that a Borel set $N \subset \mathbb{R}^{2}$ whose every point is linearly accessible by an $s$-dimensional family of lines has Hausdorff dimension at most $2 - s$.

math.CA↗
arXiv2023

Rectifiability and tangents in a rough Riemannian setting

Characterizing rectifiability of Radon measures in Euclidean space has led to fundamental contributions to geometric measure theory. Conditions involving existence of principal values of certain singular integrals \cite{mattila1995rectifiable} and the existence of densities with respect to Euclidean balls \cite{preiss1987geometry} have given rise to major breakthroughs. We study similar questions in a rough elliptic setting where Euclidean balls $B(a,r)$ are replaced by ellipses $B_{\Lambda}(a,r)$ whose eccentricity and principal axes depend on $a$. Given $\Lambda : \mathbb{R}^{n} \to GL(n,\mathbb{R})$, consider the family of ellipses $B_{\Lambda}(a,r) = a + \Lambda(a) B(0,r)$. We characterize $m$-rectifiability in terms of the almost everywhere existence of the densities $$ \theta^{m}_{\Lambda(a)}(\mu,a) = \lim_{r \downarrow 0} \frac{\mu(B_{\Lambda}(a,r))}{r^{m}} \in (0, \infty). $$ We characterize $m$-rectifiable measures in terms of the existence of the principal values-- and even under the weaker assumptions that $$ \lim_{\epsilon \downarrow 0} \int_{B_{\Lambda}(a,\epsilon R) \setminus B_{\Lambda}(a, \epsilon r)} \frac{\Lambda(a)^{-1}(y-a)}{|\Lambda(a)^{-1}(y-a)|^{m+1}} d \mu(y) = 0 \quad \forall 0 < r < R $$ when $0 < \theta^{m}_{*}(\mu,a) < \infty$ almost everywhere. We apply the second result to characterize $(n-1)$-rectifiable measures in $\mathbb{R}^{n}$ in terms of the behavior of the gradient of the single layer potential to the PDE $L_{A} u = - \textrm{div}(A \nabla u)$ under weak continuity assumptions on $A$.

math.AP↗
arXiv2023

Flat interior singularities for area almost-minimizing currents

The interior regularity of area-minimizing integral currents and semi-calibrated currents has been studied extensively in recent decades, with sharp dimension estimates established on their interior singular sets in any dimension and codimension. In stark contrast, the best result in this direction for general almost-minimizing integral currents is due to Bombieri in the 1980s, and demonstrates that the interior regular set is dense. The main results of this article show the sharpness of Bombieri's result by constructing two families of examples of area almost-minimizing integral currents whose flat singular sets contain any closed, empty interior subset $K$ of an $m$-dimensional plane in $\mathbb{R}^{m+n}$. The first family of examples are codimension one currents induced by a superposition of $C^{k,\alpha_*}$ graphs with $K$ contained in the boundary of their zero set. The second family of examples are two dimensional area almost-minimizing integral currents in $\mathbb{R}^4$ whose set of branching singularities contains $K$.

math.AP↗
arXiv2022

A study in quantitative equidistribution on the unit square

The distributional properties of the translation flow on the unit square have been considered in different fields of mathematics, including algebraic geometry and discrepancy theory. One method to quantify equidistribution is to compare the error between the actual time the translation flow spent in specific sets $E \subset [0,1]^2$ to the expected time. In this article, we prove that when $E$ is in the algebra generated by convex sets the error is of order at most $\log(T)^{1+\varepsilon}$ for all but countably many directions. Whenever the direction is badly approximable the bound can be sharpened to $\log(T)^{1/2+\varepsilon}$. The error estimates we produce are smaller than for general measurable sets as proved by Beck, while our class of examples is larger than in the work of Grepstad-Larcher who obtained the bounded remainder property for their sets. Our proof relies on the duality between local convexity of the boundary and regularity of sections of the flow.

math.DS↗
arXiv2020

A Harnack inequality for weak solutions of the Finsler $\gamma$-Laplacian

We study regularity of the Finsler $\gamma$-Laplacian, a general class of degenerate elliptic PDEs which naturally appear in anisotropic geometric problems. Precisely, given any strictly convex family of $C^{1}$-norms $\{ \rho_{x}\}$ on $\mathbb{R}^{n}$ and $\gamma > 1$, we consider the $W^{1,\gamma}(\Omega)$ solutions of the anisotropic PDE $$ \displaystyle \int_{\Omega} \left \langle \rho_{x}(Du)^{\gamma-1} (D \rho_{x})(Du), D \varphi \right \rangle = \int_{\Omega} \vec{F} \cdot D \varphi + f \varphi \qquad \forall \varphi \in W^{1,\gamma^{\prime}}_{0}(\Omega). $$ Under the mild assumption $|\xi|^{-1} \rho_{x}( \xi) \in [\nu, \Lambda]$ for all $(x,\xi) \in \Omega \times \mathbb{R}^{n}$ and some $0 < \nu \le \Lambda < \infty$ we perform a Moser iteration, verifying that sub- and super-solutions satisfy one-sided $\| \cdot \|_{\infty}$ bounds, which together imply solutions are locally bounded. When $u$ is non-negative this also implies a (weak) Harnack inequality. If $f, \vec{F} \equiv 0$ weak solutions also benefit from a strong maximum principle, and a Liouville-type theorem.

math.AP↗
arXiv2020

A geometric proof of regularity of all anisotropic minimal surfaces in $\mathbb{R}^{2}$

A set of locally finite perimeter $E \subset \mathbb{R}^{n}$ is called an anisotropic minimal surface in an open set $A$ if $\Phi(E;A) \le \Phi(F;A)$ for some surface energy $\Phi(E;A) = \int_{\partial^{*}E \cap A} \| \nu_{E}\| d \mathcal{H}^{n-1}$ and all sets of locally finite perimeter $F$ such that $E \Delta F \subset \subset A$. In this short note we provide the details of a geometric proof verifying that all anisotropic surface minimizers in $\mathbb{R}^{2}$ whose corresponding integrand $\| \cdot \|$ is strictly convex are locally disjoint unions of line segments. This demonstrates that, in the plane, strict convexity of $\| \cdot \|$ is both necessary and sufficient for regularity. The corresponding Bernstein theorem is also proven: global anisotropic minimizers $E \subset \mathbb{R}^{2}$ are half-spaces.

math.DG↗
arXiv2020

Families of well approximable measures

We provide an algorithm to approximate a finitely supported discrete measure $\mu$ by a measure $\nu_{N}$ corresponding to a set of $N$ points so that the total variation between $\mu$ and $\nu_N$ has an upper bound. As a consequence if $\mu$ is a (finite or infinitely supported) discrete probability measure on $[0,1]^{d}$ with a sufficient decay rate on the weights of each point, then $\mu$ can be approximated by $\nu_N$ with total variation, and hence star-discrepancy, bounded above by $(\log N) N^{-1}$. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure $\mu$, there exist finite sets whose star-discrepancy with respect to $\mu$ is at most $(\log N)^{d-\frac{1}{2}} N^{-1}$. Moreover we close a gap in the literature for discrepancy in the case $d=1$ showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure.

math.NT↗
arXiv2019

Menger curvatures and $C^{1,\alpha}$ rectifiability of measures

We further develop the relationship between $\beta$-numbers and discrete curvatures to provide a new proof that under weak density assumptions, finiteness of the pointwise discrete curvature $\operatorname{curv}^{\alpha}_{\mu;2}(x,r)$ at $\mu$- a.e. $x \in \mathbb{R}^{m}$ implies that $\mu$ is $C^{1,\alpha}$ $n$-rectifiable.

math.MG↗
arXiv2019

Two Phase Free Boundary Problem for Poisson Kernels

We provide a potential theoretic characterization of vanishing chord-arc domains under minimal assumptions. In particular we show that, if a domain has Ahlfors regular boundary, the oscillation of the logarithm of the interior and exterior Poisson kernels yields a great deal of geometric information about the domain. We use techniques from the classical calculus of variations, potential theory, quantitative geometric measure theory to accomplish this. One feature of this work, compared to Bortz-Hofmann PAMS 16 and Kenig-Toro Crelle 06, is that a priori we only require that the domains in question are connected.

math.CA↗
arXiv2019

Wild examples of rectifiable sets

We study the geometry of sets based on the behavior of the Jones function, $J_{E}(x) = \int_{0}^{1} \beta_{E;2}^{1}(x,r)^{2} \frac{dr}{r}$. We construct two examples of countably $1$-rectifiable sets in $\mathbb{R}^{2}$ with positive and finite $\mathcal{H}^1$-measure for which the Jones function is nowhere locally integrable. These examples satisfy different regularity properties: one is connected and one is Ahlfors regular. Both examples can be generalized to higher-dimension and co-dimension.

math.CA↗
arXiv2018

Characterizations of countably $n$-rectifiable Radon measures by higher-dimensional Menger curvatures

In the late `90s there was a flurry of activity relating $1$-rectifiable sets, boundedness of singular integral operators, the analytic capacity of a set, and the integral Menger curvature in the plane. In `99 Leger extended the results for Menger curvature to $1$-rectifiable sets in higher dimension, as well as to the codimension one case. A decade later, Lerman and Whitehouse, and later Meurer, found higher-dimensional geometrically motivated generalizations of Menger curvature that yield results about the uniform rectifiability of measures and the rectifiability of sets respectively. In this paper, we provide an extension of Meurer's work that yields a characterization of countably $n$-rectifiable measures in terms of $\sigma$-finiteness of the integral Menger curvature. We also prove that a finiteness condition on pointwise Menger curvature proves rectifiability of Radon measures. Finally, motivated by the partial converse of Meurer's work by Kolasinski we prove that under suitable density assumptions there is a comparability between pointwise-Menger curvature and the sum over scales of the centered $\beta$-numbers at a point.

math.CA↗
arXiv2015

Numerical Investigation of Metrics for Epidemic Processes on Graphs

This study develops the epidemic hitting time (EHT) metric on graphs measuring the expected time an epidemic starting at node $a$ in a fully susceptible network takes to propagate and reach node $b$. An associated EHT centrality measure is then compared to degree, betweenness, spectral, and effective resistance centrality measures through exhaustive numerical simulations on several real-world network data-sets. We find two surprising observations: first, EHT centrality is highly correlated with effective resistance centrality; second, the EHT centrality measure is much more delocalized compared to degree and spectral centrality, highlighting the role of peripheral nodes in epidemic spreading on graphs.

physics.soc-ph↗
arXiv2014

Modulus of families of walks on graphs

We introduce the notion of modulus of families of walks on graphs. We show how Beurling's famous criterion for extremality, that was formulated in the continuous case, can be interpreted on graphs as an instance of the Karush-Kuhn-Tucker conditions. We then develop an algorithm to numerically compute modulus using Beurling's criterion as our guide.

math.CO↗