Related papers: Harmonic functions with finite $p$-energy on lampl…

In this paper, we prove sharp $\ell^2$ decay estimates of nonnegative generalized subharmonic functions on graphs with positive Laplacian spectrum, which extends the result by Li and Wang on Riemannian manifolds.

We prove a global version of the classical result that $p$-harmonic functions belong to $W^{2,2}_{loc}$ for $1<p<3+\frac{2}{n-2}$. The proof relies on Cordes' matrix inequalities [7] and techniques from the work of Cianchi and Maz'ya [5,6].

Let $G$ be a finitely generated infinite group and let $p > 1$. In this paper we make a connection between the first $L^p$-cohomology space of $G$ and $p$-harmonic functions on $G$. We also describe the elements in the first…

We prove that any stationary random graph satisfying a growth condition and having positive entropy almost surely admits an infinite dimensional space of bounded harmonic functions. Applications to random infinite planar triangulations and…

We explore the relationship between convex and subharmonic functions on discrete sets. Our principal concern is to determine the setting in which a convex function is necessarily subharmonic. We initially consider the primary notions of…

In this paper we continue the investigation of the regularity of the so-called weak $\frac{n}{p}$-harmonic maps in the critical case. These are critical points of the following nonlocal energy \[ {\mathcal{L}}_s(u)=\int_{\mathbb{R}^n}| (…

We analyze the following dynamics on a connected graph $(V,E)$ with $n$ vertices. Let $V = I \bigcup B$, where the set of interior vertices $I \ne \emptyset$ is disjoint from the set of boundary vertices $B \neq \emptyset$. Given $p > 1$…

On a Riemannian surface, the energy of a map into a Riemannian manifold is a conformal invariant functional, and its critical points are the harmonic maps. Our main result is a generalization of this theorem when the starting manifold is…

We find the sharp constants $C_p$ and the sharp functions $C_p=C_p(x)$ in the inequality $$|u(x)|\leq \frac{C_p}{(1-|x|^2)^{(n-1)/p}}\|u\|_{h^p(B^n)}, u\in h^p(B^n), x\in B^n,$$ in terms of Gauss hypergeometric and Euler functions. This…

We provide a general treatment of perturbations of a class of functionals modeled on convolution energies with integrable kernel which approximate the $p$-th norm of the gradient as the kernel is scaled by letting a small parameter…

In this article we first establish the maximum principle of the antisymmetric functions for parabolic fractional $p$-equations. Then we use it and the parabolic inequalities to provide a different proof of symmetry and monotonicity for…

We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushima's ergodic theorem for the harmonic functions in the domain of the $ L^{p} $ generator. Secondly we prove analogues of Yau's and Karp's…

Given a complete doubling metric measure space $X$ that supports a $2$-Poincar\'e inequality, we approximate harmonic functions on a bounded domain $\Omega$ with a prescribed Newton-Sobolev boundary data. Our approach is based on the…

We study the equivalence between the $L^p$-parabolicity, the $L^q$-Liouville property of positive super-harmonic functions, and the existence of nonharmonic positive solutions to the following elliptic differential system \begin{equation*}…

Given a bounded planar domain $\Omega \subset \mathbb{R}^2$, we show that any singular harmonic map into the circle $\mathbb{S}^1$ corresponding to a topologically nondegenerate critical point of the renormalised energy in the sense of…

For a K\"{a}hler manifold endowed with a weighted measure $e^{-f}\,dv,$ the associated weighted Hodge Laplacian $\Delta _{f}$ maps the space of $(p,q)$-forms to itself if and only if the $(1,0)$-part of the gradient vector field $\nabla f$…

The purpose of this note is to study the connectedness at infinity of manifold by using the theory of $p$-harmonic functions. We show that if the first eigenvalue $\lambda_{1,p}$ for the $p$-Laplacian achievies its maximal value on a…

Let T be a positive closed (p,p)-current of mass 1 on a compact Kahler manifold X. Then, there exist a constant c, independent of T, and smooth positive closed (p,p)-currents Tn and Sn of mass c such that Tn-Sn converge weakly to T. We also…

We study several problems related to the $\ell^p$ boundedness of Riesz transforms for graphs endowed with so-called bounded Laplacians. Introducing a proper notion of gradient of functions on edges, we prove for $p\in(1,2]$ an $\ell^p$…

In this paper, we establish the following Liouville theorem for fractional \emph{p}-harmonic functions. {\em Assume that $u$ is a bounded solution of $$(-\lap)^s_p u(x) = 0, \;\; x \in \mathbb{R}^n,$$ with $0<s<1$ and $p \geq 2$. Then $u$…