arXiv subjects

Zhihao Lu

Publications and source records attributed to Zhihao Lu.

7 recordsLinked to original sources

arXiv2026

Global and local properties of solutions of elliptic equations with a nonlinear term involving the product of the function and its gradient

We study the global and local properties of positive solutions to the quasi-linear elliptic equation: \d u+|\nabla u|^q u^p=0,\quad x\in \O\subset \mathbb{R}^n,\nonumber where $q\ge 0$ and $p\in\mathbb{R}$. Our contributions are twofold: 1. Based on an optimal and new identity for the modulus squared of the logarithmic gradient, we establish optimal and improved Liouville theorems for global positive solutions, and generalize these findings to the framework of Riemannian manifolds. 2. Based on a newly discovered mutual control relationship of two nonlinear iterms, for all index pairs $ (p, q) $ where the Liouville theorem holds, we derive several optimal gradient estimates for local positive solutions. As a direct corollary, we obtain the corresponding Harnack inequality. These results strengthen the related conclusions in Bidaut-V\'eron--Garc\'ia-Huidobro--V\'eron \cite {BGV} from both global and local perspectives.

math.AP↗

arXiv2025

Prune4Web: DOM Tree Pruning Programming for Web Agent

Web automation employs intelligent agents to execute high-level tasks by mimicking human interactions with web interfaces. Despite the capabilities of recent Large Language Model (LLM)-based web agents, navigating complex, real-world webpages efficiently remains a significant hurdle due to the prohibitively large size of Document Object Model (DOM) structures, often ranging from 10,000 to 100,000 tokens. Existing strategies typically rely on crude DOM truncation -- risking the loss of critical information -- or employ inefficient heuristics and separate ranking models, failing to achieve an optimal balance between precision and scalability. To address these challenges, we introduce Prune4Web, a novel paradigm that shifts DOM processing from resource-intensive LLM reading to efficient programmatic pruning. Central to our approach is DOM Tree Pruning Programming, where an LLM generates executable Python scoring scripts to dynamically filter DOM elements based on semantic cues from decomposed sub-tasks. This mechanism eliminates the need for LLMs to ingest raw, massive DOMs, instead delegating traversal and scoring to lightweight, interpretable programs. This methodology achieves a 25x to 50x reduction in candidate elements for grounding, thereby facilitating precise action localization while mitigating attention dilution. Furthermore, we propose a specialized data annotation pipeline and a two-turn dialogue training strategy that jointly optimizes the Planner, Programmatic Filter, and Grounder within a unified framework. Extensive experiments demonstrate state-of-the-art performance. Notably, on our low-level grounding task, Prune4Web dramatically improves accuracy from 46.8% to 88.28%, underscoring its efficacy in real-world web automation.

cs.AI↗

arXiv2024

Liouville theorems and Harnack inequalities for Allen-Cahn type equation

We first give a logarithmic gradient estimate for positive solutions of Allen-Cahn equation on Riemannian manifolds with Ricci curvature bounded below. As its natural corallary, Harnack inequality and a Liouville theorem for classical positive solutions are obtained. Later, we consider similar estimate under integral curvature condition and generalize previous results to a class nonlinear equations which contain some classical elliptic equations such as Lane-Emden equation, static Whitehead-Newell equation and static Fisher-KPP equation. Last, we briefly generalize them to equation with gradient item under Bakry-Émery curvature condition.

math.AP↗

arXiv2024

Differential Harnack inequalities for Fisher-KPP type equations on Riemannian manifolds

We obtain almost optimal differential Harnack inequalities for a class of nonlinear parabolic equations on Riemannian manifolds with Bakry-Émery Ricci curvature bounded below, which includes the classical Fisher-KPP equation and Newell-Whitehead equation. Compared to existing research, we do not impose any additional conditions on the positive solutions. As its application, we derive some optimal Liouville properties.

math.AP↗

arXiv2023

Logarithmic gradient estimate and Universal bounds for semilinear elliptic equations revisited

We derive the complete and optimal Cheng--Yau gradient estimates and universal bounds for subcritical semilinear elliptic equations on Riemannian manifolds with (Bakry-\'{E}mery) Ricci curvature bounded below. This answers a fundamental question that has existed for a long time. As a corollary, this provides a new proof of the Gidas-Spruck classical Liouville theorem. The Harnack inequality is also obtained.

math.AP↗

arXiv2023

Gradient estimate and Universal bounds for semilinear elliptic equations on RCD$^*$(K,N) metric measure spaces

We derive logarithmic gradient estimate and universal boundedness estimate for semilinear elliptic equations on \RCD\, metric measure spaces, which contains the class of Riemannian manifolds with Ricci curvature bounded below. These estimates are applicable for equations satisfying subcritical index condition,which recover many classical results even on Euclidean spaces. In certain case, these estimates are optimal even on \RCD\,\,spaces with $K<0$. Two direct corollaries of these estimates are Harnack inequality and Liouville theorem. In addition to these estimates, we also establish fundamental relations among the universal boundedness estimate, the logarithmic gradient estimate, and Harnack inequality. Under certain and wild assumptions for the nonlinear term, we prove that these estimates are $\kappa$-equivalent on \RCDO\,spaces for any $\kappa>1$.

math.AP↗

arXiv2023

Differential Harnack inequalities for semilinear parabolic equations on Riemannian manifolds I: Bakry-Émery curvature bounded below

In this paper, we present a unified method for deriving differential Harnack inequalities for positive solutions of the semilinear parabolic equation \begin{equation*} \partial_t u=Δ_V u+H(u) \end{equation*} on complete Riemannian manifolds with Bakry-Émery curvature bounded below. This method transforms the problem of deriving differential Harnack inequalities into solving a related ODE system. As an application of this method, we obtain new and improved estimates for logarithmic-type equations and Yamabe-type equations. Moreover, under the non-negative Bakry-Émery curvature condition, we obtain complete sharp estimates for these equations. As a natural consequence of these results, we also establish sharp Harnack inequalities and Liouville-type theorems for these equations.

math.AP↗