P-harmonic Measure Research Articles

Semiregular and strongly irregular boundary points for p-harmonic functions on unbounded sets in metric spaces

The trichotomy between regular, semiregular, and strongly irregular boundary points for p-harmonic functions is obtained for unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1<p<infty . We show that these are local properties. We also deduce several characterizations of semiregular points and strongly irregular points. In particular, semiregular points are characterized by means of capacity, p-harmonic measures, removability, and semibarriers.

The Behavior at Infinity of p-Harmonic Measure in an Infinite Slab

We estimate the p-harmonic measure of the part of the boundary of an infinite slab outside a cylinder with large radius in terms of the radius.

Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular spaces

We use sphericalization to study the Dirichlet problem, Perron solutions and boundary regularity for p-harmonic functions on unbounded sets in Ahlfors regular metric spaces. Boundary regularity for the point at infinity is given special attention. In particular, we allow for several "approach directions" towards infinity and take into account the massiveness of their complements. In 2005, Llorente-Manfredi-Wu showed that the p-harmonic measure on the upper half space $R^n_+, n \ge 2$, is not subadditive on null sets when $p \neq 2$. Using their result and spherical inversion, we create similar bounded examples in the unit ball $B \subset R^n$ showing that the n-harmonic measure is not subadditive on null sets when $n \ge 3$, and neither are the p-harmonic measures in $B$ generated by certain weights depending on $p\neq 2$ and $n \ge 2$.

On p-harmonic measures in half-spaces

For all $1<p<\infty$ and $N\ge 2$ we prove that there is a constant $\alpha(p,N)>0$ such that the $p$-harmonic measure in $\R^N_+$ of a ball of radius $0 < \delta \leq 1$ in $\R^{N-1}$ is bounded above and below by a constant times $\delta ^{\alpha (p.N)}$. We provide explicit estimates for the exponent $\alpha(p,N)$

The p-Harmonic Measure of Small Axially Symmetric Sets

Using the first eigenvalue/eigenvector pair of a singular eigenvalue problem (motivated by the Dirichlet eigenvalue problem for the Laplace-Beltrami operator on a spherical cap), we define certain nonnegative p-superharmonic and p-subharmonic functions on a convex cone which are singular at the vertex and vanish on the rest of the boundary. We use these functions to give upper and lower estimates of the p-harmonic measure near the vertex of the cone as well as the p-harmonic measure of a small spherical cap.

The weak-A∞ property of harmonic and p-harmonic measures implies uniform rectifiability

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $\Omega:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$. More generally, we establish a similar result for the Riesz measure, $p$-harmonic measure, associated to the $p$-Laplace operator, $1<p<\infty$.

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 $\mathbb{R}^{n}$, $n\geq 2$. We prove that for fixed $p$ with $2<p<\infty$ there exists a Reifenberg flat domain $\Omega\subset\mathbb{R}^{n}$, $n\geq 2$, with $\mathcal{H}^{n-1}(\partial\Omega)<\infty$ and a Borel set $K\subset\partial\Omega$ such that $\mu(K)>0=\mathcal{H}^{n-1}(K)$ where $\mu$ is the p-harmonic measure associated to a positive weak solution to p-Laplace equation in $\Omega$ with continuous boundary value zero on $\partial\Omega$. We also show that there exists such a domain for which the same result holds when $p$ is fixed with $2-\eta<p<2$ for some $\eta>0$ provided that $n\geq 3$. This work is a generalization of a recent result of Azzam, Mourgoglou, and Tolsa when the measure $\mu$ is harmonic measure at $x$, $\omega=\omega^{x}$, associated to the Laplace equation, i.e when $p=2$.

Phragmén-Lindelöf Theorems and p-harmonic Measures for Sets Near Low-dimensional Hyperplanes

We prove estimates of a p-harmonic measure, p∈(n−m,∞], for sets in R n which are close to an m-dimensional hyperplane Λ⊂R n , m∈[0,n−1]. Using these estimates, we derive results of Phragmen-Lind ...

Decay of a p-harmonic measure in the plane

We study the asymptotic behaviour of a p-harmonic measure w(p), p is an element of (1, infinity], in a domain Omega subset of R-2, subject to certain regularity constraints. Our main result is that w(p) (B (w, delta) boolean AND partial derivative Omega, w(0)) approximate to delta(q) as delta -> 0(+), where q = q(v,p) is given explicitly as a function of v and p. Here, v is related to properties of Omega near w. If p = infinity, this extends to some domains in R-n. By a result due to Hirata, our result implies that the p-Green function for p is an element of (1, 2) is not quasi-symmetric in plane C-1,C-1-domains.

P-Harmonic functions with boundary data having jump discontinuities and Baernstein's problem

For p-harmonic functions on unweighted R 2 , with 1 < p < ∞ , we show that if the boundary values f has a jump at an (asymptotic) corner point z 0 , then the Perron solution Pf is asymptotically a + b arg ( z − z 0 ) + o ( | z − z 0 | ) . We use this to obtain a positive answer to Baernstein's problem on the equality of the p-harmonic measure of a union G of open arcs on the boundary of the unit disc, and the p-harmonic measure of G ¯ . We also obtain various invariance results for functions with jumps and perturbations on small sets. For p > 2 these results are new also for continuous functions. Finally we look at generalizations to R n and metric spaces.

P-Harmonic measure is not additive on null sets

When 1 < p < ∞ and p = 2 the p-harmonic measure on the boundary of the half plane R+ is not additive on null sets. In fact, there are finitely many sets E1, E2,...,Eκ in R, of p-harmonic measure zero, such that E1 ∪ E2 ∪ ... ∪ Eκ =R. Mathematics Subject Classification (2000): 31A15 (primary); 35J70, 60G46 (secondary).

On a two-phase free boundary condition for p-harmonic measures

Let Ωi⊂Rn,i∈{1,2} , be two (δ, r 0)-Reifenberg flat domains, for some 0 0, assume Ω1∩Ω2=∅ and that, for some w∈Rn and some 0 0 , δ~<δ^ , such that if δ≤δ~ , then Δ(w, r/2) is Reifenberg flat wit ...

Tug-of-war with noise: A game-theoretic view of the p-Laplacian

Fix a bounded domain Ω⊂Rd, a continuous function F:∂Ω→R, and constants ε>0 and 1<p,q<∞ with p−1+q−1=1. For each x∈Ω, let uε(x) be the value for player I of the following two-player, zero-sum game. The initial game position is x. At each stage, a fair coin is tossed, and the player who wins the toss chooses a vector v∈B̲(0,ε) to add to the game position, after which a random noise vector with mean zero and variance (q/p)|v|2 in each orthogonal direction is also added. The game ends when the game position reaches some y∈∂Ω, and player I's payoff is F(y). We show that (for sufficiently regular Ω) as ε tends to zero, the functions uε converge uniformly to the unique p-harmonic extension of F. Using a modified game (in which ε gets smaller as the game position approaches ∂Ω), we prove similar statements for general bounded domains Ω and resolutive functions F. These games and their variants interpolate between the tug-of-war games studied by Peres, Schramm, Sheffield, and Wilson [15], [16] (p=∞) and the motion-by-curvature games introduced by Spencer [17] and studied by Kohn and Serfaty [9] (p=1). They generalize the relationship between Brownian motion and the ordinary Laplacian and yield new results about p-capacity and p-harmonic measure

Note on p-Harmonic Measure

In this paper we continue our study of the dimension of a measure associated with a positive p-harmonic function which vanishes on the boundary of a quasicircle. Under various assumptions on the quasicircle, we obtain analogues of well known theorems for the dimension of harmonic measure due to Makarov.

The Dirichlet problem for p-harmonic functions on metric spaces

We study the Dirichlet problem for p-harmonic functions (and p-energy minimizers) in bounded domains in proper, pathconnected metric measure spaces equipped with a doubling measure and supporting a Poincare inequality. The Dirichlet problem has previously been solved for Sobolev type boundary data, and we extend this result and solve the problem for all continuous boundary data. We study the regularity of boundary points and prove the Kellogg property, i.e. that the set of irregular boundary points has zero p-capacity. We also construct p-capacitary, p-singular and p-harmonic measures on the boundary. We show that they are all absolutely continuous with respect to the p-capacity. For p = 2 we show that all the boundary measures are comparable and that the singular and harmonic measures coincide. We give an integral representation for the solution to the Dirichlet problem when p = 2, enabling us to extend the solvability of the problem to L-1 boundary data in this case. Moreover, we give a trace result for Newtonian functions when p = 2. Finally, we give an estimate for the Hausdorff dimension of the boundary of a bounded domain in Ahlfors Q-regular spaces. (Less)

Continuity of Sobolev Functions and Dirichletg Finite Harmonic Measures

We investigate the existence of finely continuous Sobolev functions u ∈ \(W^{1,p} (R^n ) \) which have traces \(u|_{R^{n - 1} } \) equal to the characteristic function of a bounded set E\( subset R^{n - 1} \). As an application we examine the existence of Dirichlet finite p-harmonic measures on the unit ball of Rn.

Existence of Dirichlet infinite harmonic measures on the unit disc

The primary purpose of this paper is to give an affirmative answer to a problem posed by Ohtsuka [13] whether there exists a p-harmonic measure on the unit disc in the 2-dimensional Euclidean space R2 with an infinite p-Dirichlet integral for the exponent 1 &lt; p &lt; 2.

Existence of dirichlet finite harmonic measures in nonlinear potential theory

It is shown that, although there exist no nonconstant 2-Dirichlet finite 2-harmonic measures on the unit disc, there exist nonconstant p-Dirichlet finite p-harmonic measures on the unit disc for every p with 1<p<2.