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Rectifiability, interior approximation and harmonic measure
We prove a structure theorem for any $n$-rectifiable set $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, satisfying a weak version of the lower ADR condition, and having locally finite $H^n$ ($n$-dimensional
A singular integral approach to a two phase free boundary problem
We present an alternative proof of a result of Kenig and Toro, which states that if $\Omega \subset \mathbb{R}^{n+1}$ is a two sided NTA domain, with Ahlfors-David regular boundary, and the $\log$ of
Zonal changes in renal structure and phospholipid metabolism in potassium-deficient rats.
The results indicate that potassium depletion induces early increases in the formation of cell membrane phospholipid which correlate with specific morphologic changes in different zones within the kidney.
Nonlocal self-improving properties: a functional analytic approach
A functional analytic approach to obtaining self-improving properties of solutions to linear non-local elliptic equations is presented. It yields conceptually simple and very short proofs of some
Quantitative Fatou Theorems and Uniform Rectifiability
We show that a suitable quantitative Fatou Theorem characterizes uniform rectifiability in the codimension 1 case.
Harmonic measure and approximation of uniformly rectifiable sets
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, be a uniformly rectifiable set of dimension $n$. We show $E$ that has big pieces of boundaries of a class of domains which satisfy a 2-sided corkscrew
Non-local Gehring Lemmas in Spaces of Homogeneous Type and Applications
We prove a self-improving property for reverse Hölder inequalities with non-local right-hand side. We attempt to cover all the most important situations that one encounters when studying elliptic and
Reifenberg Flatness and Oscillation of the Unit Normal Vector
We show (under mild topological assumptions) that small oscillation of the unit normal vector implies Reifenberg flatness. We then apply this observation to the study of chord-arc domains and to a