Related papers: Unique continuation on planar graphs

We consider Laplacians on $\Z^2$-periodic discrete graphs. The following results are obtained: 1) The Floquet-Bloch decomposition is constructed and basic properties are derived. 2) The estimates of the Lebesgue measure of the spectrum in…

An improvement of the Liouville theorem for discrete harmonic functions on $\mathbb{Z}^2$ is obtained. More precisely, we prove that there exists a positive constant $\varepsilon$ such that if $u$ is discrete harmonic on $\mathbb{Z}^2$ and…

In this note we give a quantitative version of the following simple observation: a discrete harmonic function on the lattice may vanish at each point of a large cube without being zero identically, at the same time there is a version of…

Add to each level of binary tree edges to make the induced graph on the level a uniform expander. It is shown that such a graph admits no non-constant bounded harmonic functions.

Let $G=(V,E)$ be a connected finite graph and $C(V)$ be the set of functions defined on $V$. Let $\Delta_p$ be the discrete $p$-Laplacian on $G$ with $p>1$ and $L=\Delta_p-k$, where $k\in C(V)$ is positive everywhere. Consider the operator…

We prove that subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some asymptotically small sets on spheres, are bounded from above everywhere. It follows that subharmonic functions of…

A Path-Complete Lyapunov Function is an algebraic criterion composed of a finite number of functions, called its pieces, and a directed, labeled graph defining Lyapunov inequalities between these pieces. It provides a stability certificate…

We prove various results connecting structural or algebraic properties of graphs and groups to conditions on their spaces of harmonic functions. In particular: we show that a group with a finitely supported symmetric measure has a…

We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs. We first show that, for locally finite graphs and a certain family of metrics, completeness of the graph implies uniqueness of these…

In the first part of the paper, we classify linear integrable (multi-dimensionally consistent) quad-equations on bipartite isoradial quad-graphs in $\mathbb C$, enjoying natural symmetries and the property that the restriction of their…

Assume that $f$ is Dunkl polyharmonic in $\mathbb{R}^n$ (i.e. $(\Delta_h)^p f=0$ for some integer $p$, where $\Delta_h$ is the Dunkl Laplacian associated to a root system $R$ and to a multiplicity function $\kappa$, defined on $R$ and…

The unique continuation on quadratic curves for harmonic functions is discussed in this paper. By using complex extension method, the conditional stability of unique continuation along quadratic curves for harmonic functions is illustrated.…

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 $u$…

We extend the classical Liouville Theorem from Laplacian to the fractional Laplacian, that is, we prove Every $\alpha$-harmonic function bounded either above or below in all of $R^n$ must be constant.

In this paper, we study the existence of extremal functions of the discrete Sobolev inequality and Hardy-Littlewood-Sobolev inequality on lattice graphs. We introduce the discrete Concentration-Compactness principle, and prove the existence…

Let $\Gamma$ be a graph equipped with a Markov operator $P$. We introduce discrete fractional Littlewood-Paley square functionals and prove their $L^p$-boundedness under various geometric assumptions on $\Gamma$.

We introduce the weighted graph Laplacian and the notion of Schr\"odinger operator on a locally finite weighted graph . Concerning essential self-adjointness, we extend Wojciechowski's and Dodziuk's results for graphs with vertex constant…

We establish unique continuation for various discrete nonlinear wave equations. For example, we show that if two solutions of the Toda lattice coincide for one lattice point in some arbitrarily small time interval, then they coincide…

Dyadic lattice graphs and their duals are commonly used as discrete approximations to the hyperbolic plane. We use them to give examples of random rooted graphs that are stationary for simple random walk, but whose duals have only a…

Unique continuation principles are fundamental properties of elliptic partial differential equations, giving conditions that guarantee that the solution to an elliptic equation must be uniformly zero. Since finite-element discretizations…