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The Brunn-Minkowski Inequality and A Minkowski Problem for Nonlinear Capacity
<p>In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of Brunn-Minkowski type for a nonlinear capacity, <inline-formula
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
Absolute continuity of harmonic measure for domains with lower regular boundaries
We study absolute continuity of harmonic measure with respect to surface measure on domains $\Omega$ that have large complements. We show that if $\Gamma\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors
Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition
Let $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (aka uniform domain), that is, a set which satisfies the interior Corkscrew and Harnack chain
Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries
Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 2$, be 1-sided NTA domain (aka uniform domain), i.e. a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume that
On the dimension of a certain measure in the plane
We study the Hausdorff dimension of a measure related to a positive weak solution of a certain partial differential equation in a simply connected domain in the plane. Our work generalizes work of
$\sigma$-finiteness of elliptic measures for quasilinear elliptic PDE in space
In this paper we study the Hausdorff dimension of a elliptic measure $\mu_{f}$ in space associated to a positive weak solution to a certain quasilinear elliptic PDE in an open subset and vanishing on
On the logarithm of the minimizing integrand for certain variational problems in two dimensions
AbstractLet f be a smooth convex homogeneous function of degreep, 1 < p < ∞, on $${\mathbb{C} \setminus \{0\}.}$$ We show that if u is a minimizer for the functional whose integrand is $${f(\nabla v
Note on an eigenvalue problem for an ODE originating from a homogeneous $p$-harmonic function
We discuss what is known about homogeneous solutions $ u $ to the p-Laplace equation, $ p $ fixed, $1 or $ u > 0 $ is p-harmonic in the cone, \[ K(\alpha) = \{ x = (x_1, \dots, x_n ) : x_1 > \cos
On the absolute continuity of p-harmonic measure and surface measure in Reifenberg flat domains
In this paper, we study the set of absolute continuity of p-harmonic measure, $\mu$, and $(n-1)-$dimensional Hausdorff measure, $\mathcal{H}^{n-1}$, on locally flat domains in $\mathbb{R}^{n}$,
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