arXiv subjects

Qingsong Gu

Publications and source records attributed to Qingsong Gu.

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, Poincaré inequality, heat-kernel bound, or stochastic completeness.

math.AP↗

arXiv2026

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

We study positive solutions of the superlinear Lane--Emden inequality \[ -Δu\ge σu^q,\qquad q>1, \] 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 \[ G_Ω\bigl(σg_Ω(o,\cdot)^q\bigr)(x) \le C g_Ω(o,x) \] for some pole $o\inΩ$. 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 $σ>0$, set $ν=σμ$. We show that if $d_ρ$ is a complete $ν$-adapted path metric and \[ \int_1^\infty \frac{r^{2q-1}}{ν(B_{d_ρ}(o,r))^{q-1}}\,dr=\infty, \] 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↗

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 \[ -Δu\ge v^p,\qquad -Δv\ge u^q, \qquad p,q>0,\quad pq>1, \] on arbitrary infinite, connected, locally finite weighted graphs. In the asymmetric case $p\ne q$, set $P=\max\{p,q\}$. If, for some root $o\in V$, \[ \sum_{n=2}^{\infty} \frac{n^{2pq+2P-1}}{μ(B(o,n))^{pq-1}}=\infty, \] 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 \[ \sum_{n=2}^{\infty} \frac{n^{2p-1}}{μ(B(o,n))^{p-1}}=\infty. \] Weighted half-line examples show that the critical logarithmic endpoint in the asymmetric result is sharp.

math.AP↗

arXiv2026

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 \[ -Δ_p u\ge σ(x)u^q \] on infinite locally finite connected weighted graphs, where $1 p-1$, $σ$ 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 \[ \int_1^\infty \frac{r^{\frac{pq}{p-1}-1}} {ν(B_ρ(o,r))^{\frac{q-p+1}{p-1}}} \dd r =\infty, \] 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↗

arXiv2025

BGD domains in p.c.f. self-similar sets II: spectral asymptotics for Laplacians

Let $K$ be a p.c.f. self-similar set equipped with a strongly recurrent Dirichlet form. Under a homogeneity assumption, for an open set $Ω\subset K$ whose boundary $\partial Ω$ is a graph-directed self-similar set, we prove that the eigenvalue counting function $ρ^Ω(x)$ of the Laplacian with Dirichlet or Neumann boundary conditions (Neumann only for connected $Ω$) has an explicit second term as $x\to +\infty$, beyond the dominant Weyl term. If $\partialΩ$ has a strong iterated structure, we establish that \begin{equation*} ρ^Ω(x)=ν(Ω)G\Big(\frac{\log x}2\Big)x^{\frac{d_S}2}+κ(\partialΩ)G_1\Big(\frac{\log x}2\Big)x^{\frac d2}+o\big(x^{\frac d2}\big), \end{equation*} where $G$ and $G_1$ are bounded periodic functions, $ν$ and $κ$ are certain reference measures, and $d_S$ and $d$ are dimension-related parameters.

math.FA↗

arXiv2024

BGD domains in p.c.f. self-similar sets I: boundary value problems for harmonic functions

We study the boundary value problems for harmonic functions on open connected subsets of post-critically finite (p.c.f.) self-similar sets, on which the Laplacian is defined through a strongly recurrent self-similar local regular Dirichlet form. For a p.c.f. self-similar set $K$, we prove that for any open connected subset $Ω\subset K$ whose "geometric" boundary is a graph-directed self-similar set, there exists a finite number of matrices called $\textit{flux transfer matrices}$ whose products generate the hitting probability from a point in $Ω$ to the "resistance" boundary $\partial Ω$. The harmonic functions on $Ω$ can be expressed by integrating functions on $\partial Ω$ against the probability measures. Furthermore, we obtain a two-sided estimate of the energy of a harmonic function in terms of its values on $\partial Ω$.

math.FA↗

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

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↗

arXiv2022

Anisotropic versions of the Brezis-Van Schaftingen-Yung approach at $s=1$ and $s=0$

In 2014, Ludwig showed the limiting behavior of the anisotropic Gagliardo $s$-seminorm of a function $f$ as $s\rightarrow 1^-$ and $s\rightarrow0^+$, which extend the results due to Bourgain-Brezis-Mironescu(BBM) and Maz'ya-Shaposhnikova(MS) respectively. Recently, Brezis, Van Schaftingen and Yung provided a different approach by replacing the strong $L^p$ norm in the Gagliardo $s$-seminorm by the weak $L^p$ quasinorm. They characterized the case for $s=1$ that complements the BBM formula. The corresponding MS formula for $s=0$ was later established by Yung and the first author. In this paper, we follow the approach of Brezis-Van Schaftingen-Yung and show the anisotropic versions of $s=1$ and $s=0$. Our result generalizes the work by Brezis, Van Schaftingen, Yung and the first author and complements the work by Ludwig.

math.FA↗

arXiv2021

$p$-energies on p.c.f. self-similar sets

We study $p$-energies on post critically finite (p.c.f.) self-similar sets for $1<p<\infty$, as limits of discrete $p$-energies on approximation graphs, extending the construction of Dirichlet forms, the $p=2$ setting. By suitably enlarging the choices of discrete $p$-energies, and employing the energy averaging method developed by Kusuoka-Zhou, we prove the existence of symmetric $p$-energies on affine nested fractals, and extend Sabot's celebrated criterion for existence and non-existence of Dirichlet forms on p.c.f. self-similar sets to the $1<p<\infty$ setting.

math.FA↗

arXiv2021

A new formula for the $L^p$ norm

Recently, Brezis, Van Schaftingen and the second author established a new formula for the $\dot{W}^{1,p}$ norm of a function in $C^{\infty}_c(\mathbb{R}^N)$. The formula was obtained by replacing the $L^p(\mathbb{R}^{2N})$ norm in the Gagliardo semi-norm for $\dot{W}^{s,p}(\mathbb{R}^N)$ with a weak-$L^p(\mathbb{R}^{2N})$ quasi-norm and setting $s = 1$. This provides a characterization of such $\dot{W}^{1,p}$ norms, which complements the celebrated Bourgain-Brezis-Mironescu (BBM) formula. In this paper, we obtain an analog for the case $s = 0$. In particular, we present a new formula for the $L^p$ norm of any function in $L^p(\mathbb{R}^N)$, which involves only the measures of suitable level sets, but no integration. This provides a characterization of the norm on $L^p(\mathbb{R}^N)$, which complements a formula by Maz'ya and Shaposhnikova. As a result, by interpolation, we obtain a new embedding of the Triebel-Lizorkin space $F^s_{p,2}(\mathbb{R}^N)$ (i.e. the Bessel potential space $(I-Δ)^{-s/2} L^p(\mathbb{R}^N)$), as well as its homogeneous counterpart $\dot{F}^s_{p,2}(\mathbb{R}^N)$, for $s \in (0,1)$, $p \in (1,\infty)$.

math.CA↗

arXiv2019

On nonexistence and existence of positive global solutions to heat equation with a potential term on Riemannian manifolds

We reinvestigate nonexistence and existence of global positive solutions to heat equation with a potential term on Riemannian manifolds. Especially, we give a very natural sharp condition only in terms of the volume of geodesic ball to obtain nonexistence results.

math.AP↗

arXiv2018

Metrics on the Sierpinski carpet by weight functions

We construct certain metrics on the Sierpinski carpet via a class of self-similar weight functions. Using these metrics and by applying known results, we obtain the two-sided sub-Gaussian heat kernel estimates of time change of the standard diffusion on the Sierpinski carpet with respect to self-similar measures. This proves a conjecture by Kigami.

math.FA↗

arXiv2018

Dirichlet forms and critical exponents on fractals

Let $B^σ_{2, \infty}$ denote the Besov space defined on a compact set $K \subset {\Bbb R}^d$ which is equipped with an $α$-regular measure $μ$. The {\it critical exponent} $σ^*$ is the supremum of the $σ$ such that $B^σ_{2, \infty} \cap C(K)$ is dense in $C(K)$. It is well-known that for many standard self-similar sets $K$, $B^{σ^*}_{2, \infty}$ are the domain of some local regular Dirichlet forms. In this paper, we explore new situations that the underlying fractal sets admit inhomogeneous resistance scalings, which yield two types of critical exponents. We will restrict our consideration on the p.c.f. sets. We first develop a technique of quotient networks to study the general theory of these critical exponents. We then construct two asymmetric p.c.f. sets, and use them to illustrate the theory and examine the function properties of the associated Besov spaces at the critical exponents; the various Dirichlet forms on these fractals will also be studied.

math.FA↗

arXiv2017

On a recursive construction of Dirichlet form on the Sierpiński gasket

Let $Γ_n$ denote the $n$-th level Sierpiński graph of the Sierpiński gasket $K$. We consider, for any given conductance $(a_0, b_0, c_0)$ on $Γ_0$, the Dirchlet form ${\mathcal E}$ on $K$ obtained from a recursive construction of compatible sequence of conductances $(a_n, b_n, c_n)$ on $Γ_n, n\geq 0$. We prove that there is a dichotomy situation: either $a_0= b_0 =c_0$ and ${\mathcal E}$ is the standard Dirichlet form, or $a_0 >b_0 =c_0$ (or the two symmetric alternatives), and ${\mathcal E}$ is a non-self-similar Dirichlet form independent of $a_0, b_0$. The second situation has also been studied in [Hattori et al 1994][Hambley et al 2002] as a one-dimensional asymptotic diffusion process on the Sierpiński gasket. For the spectral property, we give a sharp estimate of the eigenvalue distribution of the associated Laplacian, which improves a similar result in [Hambley et al 2002].

math.FA↗

arXiv2016

Lipschitz invariance of walk dimension on connected self-similar sets

Walk dimension is an important conception in analysis of fractals. In this paper we prove that the walk dimension of a connected compact set possessing an Alfors regular measure is an invariant under Lipschitz transforms. As an application, we show some generalized Sierpiński gaskets are not Lipschitz equivalent.

math.DS↗