Let $\Omega \subset {ℝ}^n,n\ge 3$, and let $p,1<p<\infty ,p\ne 2$, be given. In this paper we study the dimension of $p$-harmonic measures that arise from non-negative solutions to the $p$-Laplace equation, vanishing on a portion of $\partial \Omega$, in the setting of $\delta$-Reifenberg flat domains. We prove, for $p\ge n$, that there exists $\tilde{\delta }=\tilde{\delta }(p,n)>0$ small such that if $\Omega$ is a $\delta$-Reifenberg flat domain with $\delta <\tilde{\delta }$, then $p$-harmonic measure is concentrated on a set of $\sigma$-finite $H^{n-1}$-measure. We prove, for $p\ge n$, that for sufficiently flat Wolff snowflakes the Hausdorff dimension of $p$-harmonic measure...

This paper deals with the questions of the existence and uniqueness of a solution to the Dirichlet problem associated with the Dunkl Laplacian ${\Delta }_k$ as well as the hypoellipticity of ${\Delta }_k$ on noninvariant open sets.

Let $\mathscr{H}$ be a simplicial function space on a metric compact space $X$. Then the Choquet boundary $ChX$ of $\mathscr{H}$ is an $F_{\sigma }$-set if and only if given any bounded Baire-one function $f$ on $ChX$ there is an $\mathscr{H}$-affine bounded Baire-one function $h$ on $X$ such that $h=f$ on $ChX$. This theorem yields an answer to a problem of F. Jellett from [8] in the case of a metrizable set $X$.

We consider the general Schrödinger operator $L=div\left(A\left(x\right){\nabla }_x\right)-\mu$ on a half-space in ℝⁿ, n ≥ 3. We prove that the L-Green function G exists and is comparable to the Laplace-Green function $G_{\Delta }$ provided that μ is in some class of signed Radon measures. The result extends the one proved on the half-plane in [9] and covers the case of Schrödinger operators with potentials in the Kato class at infinity $K{ₙ}^{\infty }$ considered by Zhao and Pinchover. As an application we study the cone ${}_L\left(ℝⁿ₊\right)$ of all positive L-solutions continuously vanishing...

In this paper we deal with the stationary Navier-Stokes problem in a domain $\Omega$ with compact Lipschitz boundary $\partial \Omega$ and datum $a$ in Lebesgue spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of $\partial \Omega$, with possible countable exceptional set, provided $a$ is the sum of the gradient of a harmonic function and a sufficiently small field, with zero total flux for $\Omega$ bounded.

We study the existence of tangential boundary limits for harmonic functions in a Lipschitz domain, which belong to Orlicz-Sobolev classes. The exceptional sets appearing in this discussion are evaluated by use of Bessel-type capacities as well as Hausdorff measures.

We characterize the Hunt convolution kernels $\chi$ on ${\mathbf{R}}^n$ ($n\ge 2$) whose the Green type kernels on $D=\left\{\right(x_1,...,x_n)\in {\mathbf{R}}^n$; $x_1\&gt;0\}$, $V_{\chi }:C_{\mathbf{K}}\left(D\right)\ni f$$(\chi *f-\chi *\stackrel{}{}$

We study series expansions for harmonic functions analogous to Hartogs series for holomorphic functions. We apply them to study conjugate harmonic functions and the space of square integrable harmonic functions.