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Jiahuan Li

Publications and source records attributed to Jiahuan Li.

At least 19 recordsLinked to original sources
arXiv2026

The Higher-Dimensional Nitsche Conjecture: Sharp Bounds and Rigidity

Let $n\ge3$ and let $h:\A(r,1)\to\A(R,1)\subset\mathbb R^n$ be an onto homeomorphism with harmonic coordinate functions. We prove the sharp Nitsche bound \[ R\le R_{n,+}(r):=\frac{nr}{n-1+r^n}, \] and, when $h$ interchanges the two ends, the strictly stronger sharp bound \[ R\le R_{n,-}(r):=\frac{nr^{n-1}}{1+(n-1)r^n}. \] Both critical cases are rigid: equality forces, up to an orthogonal transformation, the corresponding end-preserving or end-reversing radial harmonic homeomorphism. No continuous extension to the closed annulus, boundary homeomorphism, boundary Jacobian, or sign condition on the Jacobian is assumed. The proof converts the nonzero degree of each interior direction map into a probability coupling and establishes a sharp contraction principle for vector measures under positive zonal kernels, using the strict concavity of spherical-cap barycenters. At either critical value, a second-order endpoint defect forces equality for a limiting transfer kernel, whose equality classification yields an orthogonal coupling graph. The remaining trace is locked by a Dirichlet-to-Neumann spectral gap in the end-preserving case and by endpoint H\"older regularity and uniform convergence of the direction maps in the end-reversing case.

math.AP↗
arXiv2026

Global Minimality and Rigidity of the Constraint Map Vortex

We consider the minimization problem \[ \min\left\{\int_{B_1}|Du|^2:\ u\in W^{1,2}(B_1;\mathbb R^n),\quad u=x\ \text{on }\partial B_1,\quad |u|\ge a\right\}, \quad 0<a<1. \] Figalli, Guerra, Kim, and Shahgholian proved that the canonical radial vortex is the unique global minimizer for $n\ge7$, and asked whether the same holds in dimensions \(3\le n \le6\). We answer this question affirmatively, thereby completing the global minimality and rigidity of the constraint map vortex in every dimension \(n\ge3\).

math.AP↗
arXiv2026

Strict Concavity of the Torsion Function for the Restricted Half-Laplacian in Bounded Convex Domains

Let $D\subset\mathbb{R}^n$, $n\ge2$, be a bounded convex domain, and let $u_D$ be the torsion function for the restricted half-Laplacian. We prove that $D^2u_D$ is negative definite at every point of $D$. The argument is based on the reflected harmonic extension in a slit domain. Quantitative Schauder estimates in slit domains yield parameter-uniform estimates for the first and second derivatives of the edge remainder; a Schur-complement calculation then determines the inertia of the extended Hessian near the slit edge. Superharmonicity of the logarithmic Hessian determinant and the Gleason--Wolff zero-set theorem exclude interior degeneracy. A method of continuity starting from the unit ball proves the result for smooth uniformly convex domains, and an exhaustion argument treats arbitrary bounded convex domains.

math.AP↗
arXiv2026

A Model-threshold Dimension Bound and Sharp Critical Ends for Smooth Singular Sets of Constant Positive $\sigma_k$-curvature Metrics

Let $k\in\mathbb N$ satisfy $1 0$ must obey \[ p\leq p_k(n), \] where $p_k$ is the model threshold determined by $\Hh^{p+1}\times\Sn^{n-p-1}$. When $k=2$ and $n=m^2$, we construct a smooth complete equality example on $\Sn^n\setminus\Sn^{(m^2-m-2)/2}$. We also prove that the strict inequality $p<p_k(n)$ holds under a finite positive linear-contact hypothesis.

math.DG↗
arXiv2026

A Brunn--Minkowski inequality and Convexity for the 2-Hessian eigenvalue in convex domains

We prove the strict log-concavity of the positive first eigenfunction \(-u\) of the \(2\)-Hessian equation and the strict $1/2$-convexity of the solution for the corresponding torsion problem in smooth bounded uniformly convex domains in $\mathbb{R}^{n}$. As applications, we establish the associated Brunn--Minkowski inequalities. We also show that this transformed-convexity phenomenon fails for \(3\)-Hessian equations by constructing, in dimension four, a smooth uniformly convex domain whose admissible zero-boundary solution has a nonconvex sublevel set.

math.AP↗
arXiv2026

Nonconvex Sublevel Sets For The Planar Translating Mean Curvature Equation

Translating solitons arise as models for type~II singularities of mean-convex mean curvature flow. We construct a smooth bounded uniformly convex domain \(\Om\Subset\R^2\) such that the zero-Dirichlet solution of the planar translating mean curvature equation has a nonconvex sublevel set. The construction is based on a corrected near-critical grim-reaper profile and explicit barriers on a long convex channel.

math.AP↗
arXiv2026

Strict Convexity for Solution of Liouville-Type Dirichlet Problems

We identify a common convexity structure for three exponential Dirichlet problems on smooth uniformly strictly convex domains: the Liouville equation $\Delta u=e^u$, the real equation $\sigma_2(D^2u)=e^{2u}$, and its complex counterpart $\sigma_2(u_{i\bar j})=e^{2u}$. In each case $u<0$ in the domain and $u=0$ on the boundary. We prove that \[ w=-\operatorname{arcosh}(e^{-u/2}) \] is strictly convex in the underlying real variables. The argument combines domain deformation, constant-rank theory, inverse-convexity estimates, radial ball models, boundary strict convexity, and local $C^2$ stability.

math.AP↗
arXiv2026

Graph-Compatible Power Concavity in Weakly Coupled Elliptic Systems

We introduce graph-compatible powers determined by the coupling graph for multicomponent weakly coupled elliptic systems and establish concavity of the associated transformed components through a concave-envelope method and a weighted viscosity comparison principle. We further establish a componentwise constant-rank theorem for the transformed Hessians, which yields strict power concavity under additional assumptions.

math.AP↗
arXiv2026

Bernstein-type theorem for stationary hypersurfaces of the Euler-Dierkes-Huisken functional

We say that a hypersurface $\Sigma \subset\mathbb{R}^{n+1}$ is $\alpha$-stationary if it is a critical point of the Euler-Dierkes-Huisken functional $\mathcal{E}_\alpha(\Sigma)=\int_\Sigma|X|^\alpha\, d\mathcal{H}^n$, introduced by Dierkes and Huisken in \cite{[DH-24]}. In this paper, we prove that every smooth, complete, connected, embedded $\alpha$-stationary hypersurface in $\mathbb{R}^{n+1}$ passing through the origin with $\alpha>0$ is a linear hyperplane.

math.DG↗
arXiv2026

Brunn--Minkowski Inequality for the First Complex $\sigma_{2}$-Hessian Eigenvalue

There are relatively few results on the convexity of solutions to complex equations. In this paper, We prove a strict real log-concavity theorem for the first eigenfunction of the complex $\sigma_{2}$-Hessian operator on smooth, bounded, real uniformly strictly convex domains in $\mathbb{C}^{n}$. As an application, we obtain a Brunn--Minkowski inequality for the first complex $\sigma_{2}$-Hessian eigenvalue. The proof combines a Bian--Guan constant-rank argument, a new inverse-convexity lemma for the compressed real Hessian, and Salani's viscosity admissible-test-function method.

math.AP↗
arXiv2026

A Brunn--Minkowski inequality for the Hessian eigenvalue in convex domain

We use the deformation methods to obtain the strictly log concavity of solution of a class Hessian equation in bounded convex domain in $\mathbb{R}^{n}$, as an application we get the Brunn--Minkowski inequality for the Hessian eigenvalue and characterize the equality case in bounded strictly convex domain in $\mathbb{R}^{n}$.

math.AP↗
arXiv2026

Coupling Heterarchical Granular Dynamics and Computational Fluid Dynamics

Granular flows in ambient fluids exhibit grain-size-dependent segregation, which is difficult to capture efficiently with existing models, especially in large-scale systems involving more than a million grains. We develop a two-way coupled framework that integrates heterarchical granular dynamics (HGD) with a fluid-fraction-weighted incompressible Navier-Stokes solver. This heterarchical granular-fluid dynamics (HGFD) model extends a previous HGD model for quasi-static deformations by introducing inertial, force-balance-driven particle velocities and consistent fluid-solid momentum exchange. The coupling between the inertial HGD and the fluid solver is performed using a staggered explicit sequential scheme and co-located Eulerian fields. The framework is evaluated against experimental data of (i) single-particle settling to verify inertial relaxation, (ii) hindered settling to reproduce concentration-dependent settling and vertical size stratification, and (iii) representative cases covering three reported segregation types to assess regime sensitivity. These results establish HGFD as an efficient and consistent approach for simulating fluid-coupled granular segregation dynamics.

cond-mat.soft↗
arXiv2026

Robust Self-Supervised Cross-Modal Super-Resolution against Real-World Misaligned Observations

Cross-modal super-resolution (SR) on real-world misaligned data is challenging, as only unlabeled low-resolution (LR) source and high-resolution (HR) guide images with complex spatial misalignment are available. Previous methods either rely on simulated training data or adopt suboptimal alignment strategies that overlook cross-modal dependencies, limiting their practical performance. To address these issues, we propose RobSelf, a self-supervised model that jointly optimizes a misalignment-aware feature translator and a content-aware reference filter online. The translator resolves unsupervised cross-modal and cross-resolution alignment via weakly-supervised, misalignment-aware translation, yielding an aligned guide feature. Guided by this feature, the filter performs reference-based discriminative self-enhancement on the source, enabling SR prediction with high resolution and high fidelity. Experiments on synthesized data and collected real-world data demonstrate that RobSelf achieves state-of-the-art performance, outperforming existing self-supervised and supervised methods. Moreover, it achieves superior efficiency, being up to 15.3$\times$ faster than prior self-supervised methods.

cs.CV↗
arXiv2026

LongCat-Flash-Thinking-2601 Technical Report

We introduce LongCat-Flash-Thinking-2601, a 560-billion-parameter open-source Mixture-of-Experts (MoE) reasoning model with superior agentic reasoning capability. LongCat-Flash-Thinking-2601 achieves state-of-the-art performance among open-source models on a wide range of agentic benchmarks, including agentic search, agentic tool use, and tool-integrated reasoning. Beyond benchmark performance, the model demonstrates strong generalization to complex tool interactions and robust behavior under noisy real-world environments. Its advanced capability stems from a unified training framework that combines domain-parallel expert training with subsequent fusion, together with an end-to-end co-design of data construction, environments, algorithms, and infrastructure spanning from pre-training to post-training. In particular, the model's strong generalization capability in complex tool-use are driven by our in-depth exploration of environment scaling and principled task construction. To optimize long-tailed, skewed generation and multi-turn agentic interactions, and to enable stable training across over 10,000 environments spanning more than 20 domains, we systematically extend our asynchronous reinforcement learning framework, DORA, for stable and efficient large-scale multi-environment training. Furthermore, recognizing that real-world tasks are inherently noisy, we conduct a systematic analysis and decomposition of real-world noise patterns, and design targeted training procedures to explicitly incorporate such imperfections into the training process, resulting in improved robustness for real-world applications. To further enhance performance on complex reasoning tasks, we introduce a Heavy Thinking mode that enables effective test-time scaling by jointly expanding reasoning depth and width through intensive parallel thinking.

cs.AI↗
arXiv2026

Unlocking Implicit Experience: Synthesizing Tool-Use Trajectories from Text

Enabling Large Language Models (LLMs) to effectively utilize tools in multi-turn interactions is essential for building capable autonomous agents. However, acquiring diverse and realistic multi-turn tool-use data remains a significant challenge. In this work, we propose a novel text-based paradigm. We observe that textual corpora naturally contain rich, multi-step problem-solving experiences, which can serve as an untapped, scalable, and authentic data source for multi-turn tool-use tasks. Based on this insight, we introduce GEM, a data synthesis pipeline that enables the generation and extraction of multi-turn tool-use trajectories from text corpora through a four-stage process: relevance filtering, workflow & tool extraction, trajectory grounding, and complexity refinement. To reduce the computational cost, we further train a specialized Trajectory Synthesizer via supervised fine-tuning. This model distills the complex generation pipeline into an efficient, end-to-end trajectory generator. Experiments demonstrate that our GEM-32B achieve a 16.5% improvement on the BFCL V3 Multi-turn benchmark. Our models partially surpass the performance of models trained on {\tau} - bench (Airline and Retail) in-domain data, highlighting the superior generalization capability derived from our text-based synthesis paradigm. Notably, our Trajectory Synthesizer matches the quality of the full pipeline while significantly reducing inference latency and costs.

cs.CL↗
arXiv2025

Efficient Context Scaling with LongCat ZigZag Attention

We introduce LongCat ZigZag Attention (LoZA), which is a sparse attention scheme designed to transform any existing full-attention models into sparse versions with rather limited compute budget. In long-context scenarios, LoZA can achieve significant speed-ups both for prefill-intensive (e.g., retrieval-augmented generation) and decode-intensive (e.g., tool-integrated reasoning) cases. Specifically, by applying LoZA to LongCat-Flash during mid-training, we serve LongCat-Flash-Exp as a long-context foundation model that can swiftly process up to 1 million tokens, enabling efficient long-term reasoning and long-horizon agentic capabilities.

cs.CL↗
arXiv2025

Lower Bound of Nodal Sets in Elliptic Homogenization and Functions with Strong Maximum Principle

In this note, we first try to prove a uniform lower bound of nodal volume in elliptic homogenization setting. This lower bound is far from optimal. But, we can prove a constant lower bound in dimension two. Motivated by the proof, we extend this results to more general settings. To be more specific, we prove that the nodal volume has a constant lower bound for all continuous functions with strong maximum principle. Our result works for general functions beyond solutions to elliptic PDEs.

math.AP↗