Matthew
Badger

We develop geometric versions of Rademacher and Calderón type differentiability theorems in two categories. A special case of our results is that for any Lipschitz or continuous $W^{1,p}$ Sobolev map $f$ from ${[0,1]}^n$ into a Euclidean space with $p>n$, the image $f\left({[0,1]}^n\right)$ has a unique tangent set (Attouch-Wets convergence) at almost every point with respect to the $n$-dimensional Hausdorff measure. In the analogous case when $f$ is a continuous $N^{1,p}$ map from ${[0,1]}^n$ into a metric space, we show that the image $f\left({[0,1]}^n\right)$ has a unique metric tangent (Gromov-Hausdorff convergence) almost everywhere. These results complement, but are distinct from, Federer's theorem on existence and uniqueness of approximate tangents of $n$-rectifiable sets in ${\mathbb{R}}^d$.

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 × time), i.e., polynomial solutions of the heat equation satisfying ${\partial }_{t}p\not\equiv 0$ and $p(\lambda x,{\lambda }^{2}t)={\lambda }^{d}p(x,t)$ for all $x\in {\mathbb{R}}^n$, $t\in \mathbb{R}$, and $\lambda >0$. When $n=1$, the number of nodal domains is classically precisely $\lceil d/2\rceil$. When $n=2$, we prove that the minimum number of nodal domains is $2$ if $d\not\equiv 0\left(\mathrm{mod}4\right)$ and is $3$ if $d\equiv 0\left(\mathrm{mod}4\right)$. When $n\ge 3$, we prove that the minimum number of nodal domains is $2$ for all $d$. Finally, the maximum number of nodal domains is $\mathrm{\Theta }\left(d^n\right)$ as $d\to \infty$ and lies between ${\lfloor d/n\rfloor }^n$ and $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ for all $n$ and $d$.

We prove that for all integers $2\le m\le d-1$, there exist doubling measures on ${\mathbb{R}}^d$ with full support that are $m$-rectifiable and purely $(m-1)$-unrectifiable in the sense of Federer (i.e., without assuming $\mu \ll {\mathcal{H}}^m$). Our construction is informed by a simple observation about square packing in the plane: $N$ axis-parallel squares of side length $s$ pack inside a square of side length $\lceil N^{1/2}\rceil s$. One consequence is that for each $m=2,3,4$ and $s<\infty$, there exist doubling measures $\mu$ on the Heisenberg group ${\mathbb{H}}^1$ and Lipschitz maps $f:E\subset {\mathbb{R}}^m\to {\mathbb{H}}^1$ such that $\mu \ll {\mathcal{H}}^s$, $f\left(E\right)$ has Hausdorff dimension $s$, and $\mu \left(f\right(E\left)\right)>0$. This is striking because ${\mathcal{H}}^m\left(f\right(E\left)\right)=0$ for every Lipschitz map $f:E\subset {\mathbb{R}}^m\to {\mathbb{H}}^1$ by a theorem of Ambrosio and Kirchheim (2000).

In Kenig and Toro's two-phase free boundary problem, one studies how the regularity of the Radon-Nikodym derivative $h$ of exterior harmonic measure with respect to interior harmonic measure on complementary NTA domains controls the geometry of their common boundary. It is now known that $\log h$ being Hölder continuous implies that pointwise the boundary has a unique blow-up, which is the zero set of a homogeneous harmonic polynomial. In this note, we give examples of domains with $\log h$ being continuous whose boundaries have points with non-unique blow-ups. The examples arise from oscillating or rotating a blow-up limit by an infinite amount, but very slowly.

We show the David-Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require its surface measure to be upper Ahlfors regular. Thus we can study absolute continuity of harmonic measure and surface measure on NTA domains of locally finite perimeter using Lipschitz approximations. A partial analogue of the F. and M. Riesz Theorem for simply connected planar domains is obtained for NTA domains in space. As a consequence, every Wolff snowflake has infinite surface measure.

We investigate the geometry of sets in Euclidean and infinite-dimensional Hilbert spaces. We establish sufficient conditions that ensure a set of points is contained in the image of a $(1/s)$-Hölder continuous map $f:[0,1]\to {\ell }^2$, with $s>1$. Our results generalize the "sufficient half" of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in ${\mathbb{R}}^N$ or ${\ell }^2$ in terms of a quadratic sum of linear approximation numbers called Jones' $\beta$-numbers. We show how to construct parameterizations of intermediate approximating curves $f_k\left(\right[0,1\left]\right)$ and find conditions in terms of tube approximations ensuring the approximating curves converge to a Hölder curve. As an application, we provide sufficient conditions to guarantee fractional rectifiability of pointwise doubling measures in ${\mathbb{R}}^N$.

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures in $n$-dimensional Euclidean space for all $n\ge 2$ in terms of positivity of the lower density and finiteness of a geometric square function, which records in an $L^2$ gauge the extent to which the measure admits approximate tangent lines along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between the measure and $1$-dimensional Hausdorff measure. We also characterize purely 1-unrectifiable Radon measures, i.e., locally finite measures that give measure zero to every finite length curve. Along the way, we develop an $L^2$ variant of Jones' traveling salesman construction.

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in ${\mathbb{R}}^n$, $n>2$. To each locally finite Borel measure $\mu$, we associate a function $t{J}_2(\mu ,x)$ which uses a weighted sum to record how closely the mass of $\mu$ is concentrated on a line in the triples of dyadic cubes containing $x$. We show that $t{J}_2(\mu ,x)<\infty$ $\mu$-a.e. is a necessary condition for $\mu$ to give full mass to a countable family of rectifiable curves. This confirms a conjecture of Peter Jones from 2000. No assumption is made on the upper Hausdorff density of the measure, allowing analysis of generic 1-rectifiable measures that are mutually singular with ${\mathcal{H}}^1$.

We identify two sufficient conditions for locally finite Borel measures on ${\mathbb{R}}^n$ to give full mass to a countable family of Lipschitz maps of ${\mathbb{R}}^m$. The first condition, extending a prior result of Pajot, is a sufficient test in terms of $L^p$ affine approximability for a locally finite Borel measure $\mu$ satisfying ${lim\; sup}_{r\downarrow 0}\mu \left(B\right(x,r\left)\right)/r^m<\infty$ at $\mu$-almost every $x$ to be $m$-rectifiable. The second condition is an assumption on the growth rate of the 1-density that ensures a locally finite Borel measure $\mu$ with ${lim}_{r\downarrow 0}\mu \left(B\right(x,r\left)\right)/r=\infty$ at $\mu$-almost every $x$ is 1-rectifiable.