arXiv subjects

Cole Jeznach

Publications and source records attributed to Cole Jeznach.

7 recordsLinked to original sources

arXiv2026

Optimal rates of decay at infinity for solutions to Schrödinger equations

We prove rates of decay at infinity for solutions to variable-coefficient Schrödinger equations of the form $-\text{div}(A \nabla u) + W \cdot \nabla u + V u = λu$ in cylinders, $\mathbb{T}^d \times \mathbb{R}^m$. We assume that $W$ and $V$ are bounded and that $λ\in \mathbb{C}$. Our rates depend on the decay of $|\nabla A|$ at infinity. In particular, we prove a range of quantitative unique continuation-type results at infinity when $|\nabla A(θ, x)| \le C (1 + |x|)^{-τ}$ for $τ\in [0,1]$. By adapting the methods in [KLP25], we construct explicit solutions to demonstrate the sharpness of our estimates for each such $τ$.

math.AP↗

arXiv2026

Singular set estimates for solutions to elliptic equations in higher co-dimension

Recent advances in quantitative unique continuation properties for solutions to uniformly elliptic, divergence form equations (with Lipschitz coefficients) has led to a good understanding of the vanishing order and size of singular and zero set of solutions. Such estimates also hold at the boundary, provided that the domain is sufficiently regular. In this work, we investigate the boundary behavior of solutions to a class of elliptic equations in the higher co-dimension setting, whose coefficients are neither uniformly elliptic, nor uniformly Lipschitz. Despite these challenges, we are still able to show analogous estimates on the singular set of such solutions near the boundary. Our main technical advance is a variant of the Cheeger-Naber-Valtorta quantitative stratification scheme using cones instead of planes.

math.AP↗

arXiv2025

Elliptic unique continuation below the Lipschitz threshold

In this article, we investigate unique continuation principles for solutions $u$ of uniformly elliptic equations of the form $-\mathrm{div}(A \nabla u) = 0$ when $A$ is less regular than Lipschitz. For general matrices $A$, we prove that strong unique continuation holds provided that $A$ has modulus of continuity $ω$ satisfying the Osgood condition $\int_0^1 ω(t)^{-1}dt = \infty$, plus some other mild hypotheses. Along with the counterexamples of Mandache, this shows that the sharp condition on $A$ that guarantees unique continuation is essentially that $A$ is log-Lipschitz.

math.AP↗

arXiv2025

On the number of nodal domains of homogeneous caloric polynomials

We investigate the minimum and maximum number of nodal domains across all time-dependent homogeneous caloric polynomials of degree $d$ in $\mathbb{R}^{n}\times\mathbb{R}$ (space $\times$ time), i.e., polynomial solutions of the heat equation satisfying $\partial_t p\not\equiv 0$ and $$p(λx, λ^2 t) = λ^d p(x,t)\quad\text{for all $x \in \mathbb{R}^n$, $t \in \mathbb{R}$, and $λ> 0$.}$$ When $n=1$, it is classically known that the number of nodal domains is precisely $2\lceil d/2\rceil$. When $n=2$, we prove that the minimum number of nodal domains is 2 if $d\not \equiv 0\pmod 4$ and is 3 if $d\equiv 0\pmod 4$. When $n\geq 3$, we prove that the minimum number of nodal domains is $2$ for all $d$. Finally, we show that the maximum number of nodal domains is $Θ(d^n)$ as $d\rightarrow\infty$ and lies between $\lfloor \frac{d}{n}\rfloor^n$ and $\binom{n+d}{n}$ for all $n$ and $d$. As an application and motivation for counting nodal domains, we confirm existence of the singular strata in Mourgoglou and Puliatti's two-phase free boundary regularity theorem for caloric measure.

math.AP↗

arXiv2024

Small constant uniform rectifiability

We provide several equivalent characterizations of locally flat, $d$-Ahlfors regular, uniformly rectifiable sets $E$ in $\mathbb{R}^n$ with density close to $1$ for any dimension $d \in \mathbb{N}$ with $1 \le d \le n-1$. In particular, we show that when $E$ is Reifenberg flat with small constant and has Ahlfors regularity constant close to $1$, then the Tolsa alpha coefficients associated to $E$ satisfy a small constant Carleson measure estimate. This estimate is new, even when $d = n-1$, and gives a new characterization of chord-arc domains with small constant.

math.CA↗

arXiv2023

Cantor sets with absolutely continuous harmonic measure

We construct Ahlfors regular Cantor sets $K$ of small dimension in the plane, such that the Hausdorff measure on $K$ is equivalent to the harmonic measure associated to its complement. In particular the Green function in $R^2 \backslash K$ satisfies $G^p (x) \simeq \mathrm{dist} (x, K)^δ$ whenever $\mathrm{dist} (x, K) \le 1$ and $p$ is far from $K$.

math.AP↗

arXiv2022

Non-local distance functions and geometric regularity

We establish the equivalence between the regularity (rectifiability) of sets and suitable estimates on the oscillation of the gradient for smooth non-local distance functions. A prototypical example of such a distance was introduced, as part of a larger PDE theory, by Guy David, Joseph Feneuil, and the third author. The results apply to all dimensions and co-dimensions, require no underlying topological assumptions, and provide a surprisingly rich class of analytic characterizations of rectifiability.

math.CA↗