arXiv subjects

Xueping Huang

Publications and source records attributed to Xueping Huang.

16 recordsLinked to original sources

arXiv2026

Flow Decomposition and Sharp Integral Fujita Criteria on Weighted Graphs

We study the Fujita phenomenon for semilinear heat inequalities generated by variable-speed Laplacians on infinite weighted graphs. Assuming that the graph carries a proper adapted path metric, we establish an integral volume-growth criterion forcing every nonnegative global classical supersolution on the open cylinder $(0,\infty)\times V$ to vanish, without assuming an initial value or trace. We prove that a nontrivial supersolution of this kind exists if and only if the equation has a positive global Cauchy solution for some nonzero point-source datum. A complementary heat-kernel construction gives global Cauchy solutions for all sufficiently small point-source data when the same volume integral converges and a matching anchored heat-kernel upper bound is available. This proves sharpness on integer lattices and on a family of logarithmically perturbed weighted half-lines; in the latter examples, even the exponent of an iterated logarithm can determine the existence--nonexistence alternative. A finer nonexistence criterion couples intrinsic volume growth with the capacity of intrinsic annuli. Its proof combines parabolic testing, a Laplace--resolvent reduction, and a pathwise decomposition of resolvent currents. The nonexistence results require no volume-doubling property, Poincar\'e inequality, heat-kernel bound, or stochastic completeness.

math.AP↗

arXiv2026

A sharp integral criterion for the Lane--Emden system of inequalities on weighted graphs

We establish a sharp integral nonexistence criterion for the Lane--Emden system of inequalities \[ -\Delta u\ge v^p,\qquad -\Delta v\ge u^q, \qquad p,q>0,\quad pq>1, \] on arbitrary infinite, connected, locally finite weighted graphs. In the asymmetric case $p\ne q$, set $P=\max\{p,q\}$. If, for some root $o\in V$, \[ \sum_{n=2}^{\infty} \frac{n^{2pq+2P-1}}{\mu(B(o,n))^{pq-1}}=\infty, \] then every nonnegative solution $(u,v)$ satisfies $u\equiv v\equiv0$. The proof combines flow decomposition of the finite Green current with nonlinear testing. In the symmetric case $p=q>1$, the Liouville problem reduces, via the sum $u+v$, to the scalar criterion \[ \sum_{n=2}^{\infty} \frac{n^{2p-1}}{\mu(B(o,n))^{p-1}}=\infty. \] Weighted half-line examples show that the critical logarithmic endpoint in the asymmetric result is sharp.

math.AP↗

arXiv2026

Nonnegative Bakry--\'Emery Curvature on Bounded-Degree Graphs Implies Volume Doubling and Poincar\'e Inequalities

We prove that every connected simple graph of bounded degree satisfying the classical dimension-free Bakry--\'Emery condition $\mathrm{CD}(0,\infty)$ for the unnormalised Laplacian is volume doubling and supports, at all integer graph scales, a scale-invariant $L^2$-Poincar\'e inequality with dilation two, with constants depending only on the maximum degree. This settles the polynomial-growth conjecture of Cushing, Liu, and Peyerimhoff in a stronger form. The main novelty is a dimension-free adaptation of the graph-theoretic modified nonlinear heat-flow method introduced by M\"unch and extended to infinite weighted graphs by Pajot and Russ: point-mass consequences of $\Gamma_2\geq0$ and positive-resolvent smoothing replace any global $\mathrm{CD}(0,n)$ reduction, while diffusive exit-time control and finite-volume localisation yield the Poincar\'e inequality.

math.DG↗

arXiv2026

On Brezis Open Problem 3.1

Let $B_1$ be the unit disk in ${\mathbb R}^2$. We consider the harmonic map equation $$ -\Delta u=|\nabla u|^2u,$$ subject to the Dirichlet boundary condition $ u(e^{i\theta})=(R\cos\theta,R\sin\theta,\sqrt{1-R^2}):=g_R$, where $0<R<1$ and $u: B_1\to {\mathbb S}^2$ is understood in the weak harmonic-map sense. In 1983, Brezis and Coron proved the existence of two explicit solutions of this nonlinear Dirichlet problem and showed that they are the unique minimizers in their respective relative homotopy classes. In this paper, we resolve a long-standing open question originally posed in their work, later posed as Open Problem 3.1 in Brezis Favorite Open Problems List. Specifically, we prove that these two explicit maps are the only weak harmonic maps with boundary trace $g_{R}$, thereby providing a definitive affirmative answer to Brezis open problem. The proof is based on a boundary rigidity argument. An auxiliary potential $X$ associated with $u$, the Pohozaev identity for the Hopf differential, and the planar isoperimetric inequality imply $$|u_r|\equiv R, \qquad u_r\cdot u_\theta\equiv0 \qquad\text{on }\partial B_1. $$ Thus the Hopf differential vanishes on the boundary and hence, by holomorphicity, on the whole disk. The problem is then reduced to the conformal case, where a stereographic-coordinate classification gives exactly the two Brezis--Coron maps.

math.AP↗

arXiv2026

A Volume-Growth Criterion for the p-Laplace Inequality on Weighted Graphs

We prove a nonexistence result for nonnegative solutions of the quasi-linear elliptic inequality \[ -\Delta_p u\ge \sigma(x)u^q \] on infinite locally finite connected weighted graphs, where $1 p-1$, $\sigma$ is a nonnegative Radon measure. Under the non-$p$-parabolic setting, we show that every nonnegative solution is identically zero, provided the volume of intrinsic balls satisfy \[ \int_1^\infty \frac{r^{\frac{pq}{p-1}-1}} {\nu(B_\rho(o,r))^{\frac{q-p+1}{p-1}}} \dd r =\infty, \] This criterion recovers the known sharp pointwise critical volume-growth threshold and is strictly more flexible, since it allows irregular growth and does not require uniform upper bounds at every large radius. The proof adapts the finite-network current method to the $p$-Laplace setting, combining a path decomposition with one-dimensional Hardy estimates, $p$-parallel-sum bounds across metric cuts, and the global $p$-Green function furnished by non-$p$-parabolicity.

math.AP↗

arXiv2026

The Ekeland--Nirenberg Variational Problem:A Sharp Positivity Threshold and Extensions

We study the Ekeland--Nirenberg variational problem in the two-dimensional diagonal family \[ J_{a,c,d}(u)=\int_{\Rp^2}\bigl(u_{xy}^2+a u_x^2+c u_y^2+d u^2\bigr)\dd x\dd y, \qquad a,c,d>0, \] under the constraint $u(0,0)=1$. If $u_{a,c,d}$ is the unique minimizer and $K_{a,c,d}$ is its cosine kernel, we prove the sharp classification \[ K_{a,c,d}>0 \hbox{ on } \Rp^2\quad\Longleftrightarrow\quad u_{a,c,d}>0 \hbox{ on } \Rp^2\quad\Longleftrightarrow\quad d\le ac . \] Thus every supercritical triple $d>ac$ produces sign change. We also prove local sign-change stability under small two-dimensional non-diagonal perturbations and a sharp product-type $n$-dimensional diagonal threshold. The domain and evolution results are stated in precise auxiliary settings: a free-boundary capacity formulation for domains and a selected decaying branch of the second-order evolution equation.

math.AP↗

arXiv2026

Flow Decomposition, Green Testing, and Lane--Emden Inequalities on Weighted Graphs

We study positive solutions of the superlinear Lane--Emden inequality \[ -\Delta u\ge \sigma u^q,\qquad q>1, \] on infinite locally finite weighted graphs and connected domains. When the Dirichlet Green function is finite, the existence of a positive solution is equivalent to \[ G_\Omega\bigl(\sigma g_\Omega(o,\cdot)^q\bigr)(x) \le C g_\Omega(o,x) \] for some pole $o\in\Omega$. Under Green function estimates, this yields sharp existence criteria and the Serrin-type exponents on $\mathbb Z^d$ and orthant domains. For nonexistence, the principal method is flow decomposition. Its basic estimate bounds Green energy from below in terms of the relative capacities of intrinsic balls. %It yields annular conductance, capacity-to-infinity, and Nash--Williams cut-resistance criteria. For $\sigma>0$, set $\nu=\sigma\mu$. We show that if $d_\rho$ is a complete $\nu$-adapted path metric and \[ \int_1^\infty \frac{r^{2q-1}}{\nu(B_{d_\rho}(o,r))^{q-1}}\,dr=\infty, \] then every nonnegative solution is identically zero. The proof combines a flow decomposition of the acyclic Green current, a pathwise Hardy estimate, and a relative capacity estimate. It requires none of (VD), (PI), (P$_0$), or the (3G) condition.

math.AP↗

arXiv2023

Sharp criteria for nonlocal elliptic inequalities on manifolds

Let $M$ be a complete non-compact Riemannian manifold and $σ$ be a Radon measure on $M$, we study the existence and non-existence of positive solutions to a nonlocal elliptic inequality \begin{equation*} (-Δ)^α u\geq u^{q}σ\quad \text{in}\,\,M, \end{equation*} with $q>1$. When the Green function $G^{(α)}$ of the fractional Laplacian $(-Δ)^α$ exists and satisfies the quasi-metric property, we obtain necessary and sufficient criteria for existence of positive solutions. In particular, explicit conditions in terms of volume growth and the growth of $σ$ are given, when $M$ admits Li-Yau Gaussian type heat kernel estimates.

math.AP↗

arXiv2022

Bounding the diameter and eigenvalues of amply regular graphs via Lin-Lu-Yau curvature

An amply regular graph is a regular graph such that any two adjacent vertices have $\alpha$ common neighbors and any two vertices with distance $2$ have $\beta$ common neighbors. We prove a sharp lower bound estimate for the Lin--Lu--Yau curvature of any amply regular graph with girth $3$ and $\beta>\alpha$. The proof involves new ideas relating discrete Ricci curvature with local matching properties: This includes a novel construction of a regular bipartite graph from the local structure and related distance estimates. As a consequence, we obtain sharp diameter and eigenvalue bounds for amply regular graphs.

math.CO↗

arXiv2022

Superlinear elliptic inequalities on weighted graphs

Let $(V,μ)$ be an infinite, connected, locally finite weighted graph. We study the problem of existence or non-existence of positive solutions to a semi-linear elliptic inequality \begin{equation*} Δu+u^σ\leq0\quad \text{in}\,\,V, \end{equation*} where $Δ$ is the standard graph Laplacian on $V$ and $σ>0$. For $σ\in(0,1]$, the inequality admits no nontrivial positive solution. For $σ>1$, assuming condition \textbf{($p_0$)} on $(V,μ)$, we obtain a sharp condition for nonexistence of positive solutions in terms of the volume growth of the graph, that is \begin{equation*} μ(o,n)\lesssim n^{\frac{2σ}{σ-1}}(\ln n)^{\frac{1}{σ-1}} \end{equation*} for some $o\in V$ and all large enough $n$. For any $\varepsilon>0$, we can construct an example on a homogeneous tree $\mathbb T_N$ with $μ(o,n)\approx n^{\frac{2σ}{σ-1}}(\ln n)^{\frac{1}{σ-1}+\varepsilon}$, and a solution to the inequality on $(\mathbb T_N,μ)$ to illustrate the sharpness of $\frac{2σ}{σ-1}$ and $\frac{1}{σ-1}$.

math.AP↗

arXiv2018

On the uniqueness class, stochastic completeness and volume growth for graphs

In this note we prove an optimal volume growth condition for stochastic completeness of graphs under very mild assumptions. This is realized by proving a uniqueness class criterion for the heat equation which is an analogue to a corresponding result of Grigor'yan on manifolds. This uniqueness class criterion is shown to hold for graphs that we call globally local, i.e., graphs where we control the jump size far outside. The transfer from general graphs to globally local graphs is then carried out via so called refinements.

math.MG↗

arXiv2016

Diffusion on Delone sets

We consider graphs associated to Delone sets in Euclidean space. Such graphs arise in various ways from tilings. Here, we provide a unified framework. In this context, we study the associated Laplace operators and show Gaussian heat kernel bounds for their semigroups. These results apply to both metric and discrete graphs.

math.SP↗

arXiv2013

A note on self-adjoint extensions of the Laplacian on weighted graphs

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 extensions. Moreover, in the case when the graph is not metrically complete and the Cauchy boundary has finite capacity, we characterize the uniqueness of the Markovian extensions.

math.FA↗

arXiv2013

Upper escape rate of Markov chains on weighted graphs

We obtain an upper escape rate function for a continuous time minimal symmetric Markov chain, defined on a locally finite weighted graph. This upper rate function is given in terms of volume growth with respect to an adapted path metric and has the same form as the manifold setting. Our approach also gives a slightly more restrictive form of Folz's theorem on conservativeness as a consequence.

math.PR↗

arXiv2012

A note on the volume growth criterion for stochastic completeness of weighted graphs

We generalize the weak Omori-Yau maximum principle to the setting of strongly local Dirichlet forms. As an application, we obtain an analytic approach to compare the stochastic completeness of a weighted graph with that of an associated metric graph. This comparison result played an essential role in the volume growth criterion of Folz \cite{FOLZSC}, who first proved it via a probabilistic approach. We also give an alternative analytic proof based on a criterion in Fukushima, Oshima, and Takeda \cite{FOT}.

math.PR↗

arXiv2010

Stochastic incompleteness for graphs and weak Omori-Yau maximum principle

We prove an analogue of the weak Omori-Yau maximum principle and Khas'minskii's criterion for graphs in the general setting of Keller and Lenz. Our approach naturally gives the stability of stochastic incompleteness under certain surgeries of graphs. It allows to develop a unified approach to all known criteria of stochastic completeness/incompleteness, as well as to obtain new criteria.

math.SP↗