Quantitative Regularity for p-Minimizing Maps Through a Reifenberg Theorem

In this article we extend to arbitrary p-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case $$p=2$$ p = 2 . We first show that the set of singular points of such a map can be quantitatively stratified: we classify singular points based on the number of almost-symmetries of the map around them, as done in Cheeger and Naber (Commun Pure Appl Math 66(6): 965–990, 2013). Then, adapting the work of Naber and Valtorta (Ann Math (2) 185(1): 131–227, 2017), we apply a Reifenberg-type Theorem to each quantitative stratum; through this, we achieve an upper bound on the Minkowski content of the singular set, and we prove it is k-rectifiable for a k which only depends on p and the dimension of the domain.

The local structure of the free boundary in the fractional obstacle problem

Building upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null {{\mathcal{H}}^{n-1}} measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] and therefore we retrieve the same results:(i)local finiteness of the {(n-1)}-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure),(ii){{\mathcal{H}}^{n-1}}-rectifiability of the free boundary,(iii)classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most {(n-2)} in the free boundary.Instead, if {\varphi\in C^{k+1}(\mathbb{R}^{n})}, {k\geq 2}, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than {k+1}.

Dimensions of Two-Valued Sets via Imaginary Chaos

Two-valued sets are local sets of the 2D Gaussian free field (GFF) that can be thought of as representing all points of the domain that may be connected to the boundary by a curve on which the GFF takes values only in $[-a,b]$. Two-valued sets exist whenever $a+b\geq 2\lambda ,$ where $\lambda$ depends explicitly on the normalization of the GFF. We prove that the almost sure Hausdorff dimension of the two-valued set ${\mathbb{A}}_{-a,b}$ equals $d=2-2\lambda ^2/(a+b)^2$. For the key two-point estimate needed to give the lower bound on dimension, we use the real part of a “vertex field” built from the purely imaginary Gaussian multiplicative chaos. We also construct a non-trivial $d$-dimensional measure supported on ${\mathbb{A}}_{-a,b}$ and discuss its relation with the $d$-dimensional conformal Minkowski content of ${\mathbb{A}}_{-a,b}$.

The Mean Minkowski Content of Homogeneous Random Fractals

Homogeneous random fractals form a probabilistic generalisation of self-similar sets with more dependencies than in random recursive constructions. Under the Uniform Strong Open Set Condition we show that the mean D-dimensional (average) Minkowski content is positive and finite, where the mean Minkowski dimension D is, in general, greater than its almost sure variant. Moreover, an integral representation extending that from the special deterministic case is derived.

A general formula for the anisotropic outer Minkowski content of a set

We generalize to the anisotropic case some classical and recent results on the (n – 1)-Minkowski content of rectifiable sets in ℝn, and on the outer Minkowski content of subsets of ℝn. In particular, a general formula for the anisotropic outer Minkowski content is provided; it applies to a wide class of sets that are stable under finite unions.