# $\sigma$-finiteness of elliptic measures for quasilinear elliptic PDE in space

@article{Akman2015sigmafinitenessOE,
title={\$\sigma\$-finiteness of elliptic measures for quasilinear elliptic PDE in space},
author={Murat Akman and Johnny M. Lewis and Andrew Vogel},
journal={arXiv: Analysis of PDEs},
year={2015}
}
• Published 23 September 2015
• Mathematics
• arXiv: Analysis of PDEs
5 Citations

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## References

SHOWING 1-10 OF 29 REFERENCES
Hausdorff dimension and $\sigma$ finiteness of $p-$harmonic measures in space when $p\geq n$
• Mathematics
• 2013
In this paper we study a p harmonic measure, associated with a positive p harmonic function \hat{u} defined in an open set O, subset of R^n, and vanishing on a portion \Gamma of boundary of O. If p>n
On the logarithm of the minimizing integrand for certain variational problems in two dimensions
• Mathematics
• 2012
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
On the Hausdorff dimension of harmonic measure in higher dimension
Assume A=I~n\E a domain in F, n, where E is a compact set. Denote o~(A,A,x) the harmonic measure for A of A, evaluated at x ~ R d. According to 0ksendal 's theorem [O], o E= o(A, . ,x) is singular
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
Hausdorff dimension of harmonic measures in the plane
• Mathematics
• 1988
for all continuous u: aQ--*R, where /~ is the Perron solution of the Dirichlet problem with boundary values u. We must assume here that E has positive capacity, but not that 92 is regular for the
p Harmonic Measure in Simply Connected Domains
• Mathematics
• 2009
We extend to all planar simply connected domains Makarov-type results about the Hausdorff dimension of $p$-harmonic measure pioneered by Lewis and Bennewitz in the context of quasidisks. The key to
On the dimension of p-harmonic measure in space
• Mathematics
• 2013
Let R n , n 3, and let p, 1 0 small such that if is a -Reifenberg flat domain with < ˜ , then p-harmonic measure is concentrated on a set of -finite H n 1 measure. The situation is more interesting
Measure theory and fine properties of functions
GENERAL MEASURE THEORY Measures and Measurable Functions Lusin's and Egoroff's Theorems Integrals and Limit Theorems Product Measures, Fubini's Theorem, Lebesgue Measure Covering Theorems
Nonlinear Potential Theory of Degenerate Elliptic Equations
• Mathematics
• 1993
Introduction. 1: Weighted Sobolev spaces. 2: Capacity. 3: Supersolutions and the obstacle problem. 4: Refined Sobolev spaces. 5: Variational integrals. 6: A-harmonic functions. 7: A superharmonic