Searcharxiv⌕ Search

arXiv subjects

Panrui Ni

Publications and source records attributed to Panrui Ni.

15 recordsLinked to original sources
arXiv2026

Sharp convergence rates for the vanishing discount problem with hyperbolic Aubry sets

Let $H\in C^2(T^*M)$ be a Tonelli Hamiltonian on a closed connected manifold and let $u_\lambda$ solve \[ \lambda u_\lambda+H(x,Du_\lambda)=c(H)\qquad\text{in }M. \] We study the convergence rate of $u_\lambda$ to the selected critical solution $u_0$. Assume that the lifted Aubry set is a finite union $\widetilde{A}=\Gamma_1\sqcup\cdots\sqcup\Gamma_N$, where each $\Gamma_i$ is either a hyperbolic equilibrium or a periodic orbit hyperbolic in the critical energy level. We prove \[ -C\lambda\le u_\lambda-u_0\le C\lambda|\log\lambda|. \] Let $\mu_i$ be the projected Mather measure associated with $\Gamma_i$. If \[ \int_M u_0\,\mathrm d\mu_i=0\qquad \text{for every }i, \] then \[ \|u_\lambda-u_0\|_\infty\le C\lambda. \] In particular, if the lifted Aubry set consists of a single hyperbolic equilibrium or a single hyperbolic periodic orbit, the convergence rate is $O(\lambda)$. We give examples showing that both convergence rates $O(\lambda)$ and $O(\lambda|\log\lambda|)$ are optimal. Without hyperbolicity, finite-order degenerate examples give lower bounds of order $\lambda^{1/(2r-1)}$ with $r\ge2$. We also construct infinite-order degenerate examples with arbitrarily slow convergence. Taken together, these results provide, to our knowledge, the first systematic quantitative theory for the vanishing discount problem in the Tonelli setting.

math.AP↗
arXiv2026

Optimal semiconcavity with fractional modulus for Hamilton-Jacobi equations with Neumann boundary conditions

We study the semiconcavity property of viscosity solutions to Hamilton--Jacobi equations with Neumann boundary conditions. Unlike the state-constraint case, minimizing trajectories associated with the Neumann problem may fail to be $C^1$, so the classical approach based on the regularity of minimizers is no longer available. To overcome this difficulty, we introduce a comparison argument between the constrained action associated with the Skorokhod problem and the unconstrained action, avoiding any use of higher regularity of reflected minimizing trajectories. Under a structural decomposition assumption on the Hamiltonian at the boundary, we establish the estimate \[u(x+h,t+\sigma)+u(x-h,t-\sigma)-2u(x,t)\leq C(|h|+\sigma)^{\frac{3}{2}}.\] An explicit example shows that the power $3/2$ in this estimate cannot be improved.

math.AP↗
arXiv2026

A PDE formulation of Lyapunov stability for contact-type Hamilton-Jacobi equations

We study the Lyapunov stability of stationary solutions to contact-type Hamilton-Jacobi equations on a compact manifold. Previous works typically assume $C^3$ Tonelli Hamiltonians and characterize stability in terms of Mather measures. In this paper, we consider continuous, convex and coercive Hamiltonians and establish verifiable PDE-type criteria for both stability and instability. In particular, the dynamical conditions involving Mather measures are replaced by conditions expressed in terms of the critical value of the Hamiltonian and viscosity subsolutions. This provides a PDE-based framework for stability analysis and reveals connections with various asymptotic behaviors of viscosity solutions.

math.AP↗
arXiv2026

Static class-guided selection of elementary solutions in non-monotone vanishing discount problems

We study a generalized vanishing discount problem for Hamilton--Jacobi equations, removing the standard monotonicity assumption, either in a global sense or when integrated against all Mather measures. Specifically, we consider \[ \lambda a(x)u(x)+H(x,Du(x))-A\lambda=c_0, \] with a suitably chosen constant $A>0$. By appropriately changing the signs of the function $a(x)$ on different static classes associated with $H$, we show that the maximal viscosity solution converges uniformly as $\lambda\to 0^+$ and that all elementary solutions of the stationary equation \[ H(x,Du(x))=c_0 \] can be selected as limits. This provides the first result for selecting multiple viscosity solutions in vanishing discount problems beyond the usual monotonicity and integral assumptions, as long as $a(x)$ is positive on one static class. Our results highlight the crucial role of static classes in controlling the asymptotic behavior of viscosity solutions. Previously, under usual monotonicity assumptions, only a single solution could be selected (as discussed in \cite{GL}), whereas our approach allows controlled selection of multiple solutions via static class-guided discount coefficients.

math.AP↗
arXiv2025

Quantitative homogenization of first-order ODEs

This paper investigates the quantitative homogenization of first-order ODEs. For single-scale scalar ODEs, we obtain a sharp $O(\varepsilon)$ convergence rate and characterize the effective constant. In the multi-scale setting, our results match those of \cite{IM} for long times but improve the short-time error to $O(\varepsilon)$. We also initiate the study of quasi-periodic homogenization in this context. The scalar framework is further extended to higher dimensions under a boundedness assumption on trajectories. For weakly coupled systems with fast switching rates, we obtain for the first time a convergence rate of order $O(\varepsilon)$. These results have applications to linear transport equations and broader connections to PDEs and gradient systems.

math.CA↗
arXiv2025

Quantitative homogenization of convex Hamilton-Jacobi equations with $u/\varepsilon$-periodic Hamiltonians

Here, we study quantitative homogenization of first-order convex Hamilton-Jacobi equations with $(u/\varepsilon)$-periodic Hamiltonians which typically appear in dislocation dynamics. Firstly, we establish the optimal convergence rate by using the inherent fundamental solution and the implicit variational principle of Hamilton dynamics with their Hamiltonian depending on the unknown. Secondly, under additional growth assumptions on the Hamiltonian, we establish global H\"older regularity for both the solutions and the correctors, serving as a notable application of our quantitative homogenization theory.

math.AP↗
arXiv2025

Quantitative homogenization of convex Hamilton-Jacobi equations with Neumann type boundary conditions

We study the periodic homogenization for convex Hamilton-Jacobi equations on perforated domains under the Neumann type boundary conditions. We consider two types of conditions, the oblique derivative boundary condition and the prescribed contact angle boundary condition, which is important in the front propagation. We first establish a new representation formula for the solution by using the Skorokhod problem and modified Lagrangians. By using this formula essentially, we prove the sub and superadditivity properties of the extended metric functions, which will be applied to obtain the optimal convergence rate $O(\varepsilon)$ for homogenization of Neumann type problems.

math.AP↗
arXiv2024

Convergence/divergence phenomena in the vanishing discount limit of Hamilton-Jacobi equations

We study the asymptotic behavior of solutions of an equation of the form \begin{equation}\label{abs}\tag{*} G\big(x, D_x u,\lambda u(x)\big) = c_0\qquad\hbox{in $M$} \end{equation} on a closed Riemannian manifold $M$, where $G\in C(T^*M\times\mathbb{R})$ is convex and superlinear in the gradient variable, is globally Lipschitz but not monotone in the last argument, and $c_0$ is the critical constant associated with the Hamiltonian $H:=G(\cdot,\cdot,0)$. By assuming that $\partial_u G(\cdot,\cdot,0)$ satisfies a positivity condition of integral type on the Mather set of $H$, we prove that any equi-bounded family of solutions of \eqref{abs} uniformly converges to a distinguished critical solution $u_0$ as $\lambda \to 0^+$. We furthermore show that any other possible family of solutions uniformly diverges to $+\infty$ or $-\infty$. We then look into the linear case $G(x,p,u):=a(x)u + H(x,p)$ and prove that the family $(u_\lambda)_{\lambda \in (0,\lambda_0)}$ of maximal solutions to \eqref{abs} is well defined and equi-bounded for $\lambda_0>0$ small enough. When $a$ changes sign and enjoys a stronger localized positivity assumption, we show that equation \eqref{abs} does admit other solutions too, and that they all uniformly diverge to $-\infty$ as $\lambda \to 0^+$. This is the first time that converging and diverging families of solutions are shown to coexist in such a generality.

math.AP↗
arXiv2024

Nonlinear and degenerate discounted approximation in discrete weak KAM theory

In this paper, we introduce a discrete version of the nonlinear implicit Lax-Oleinik operator. We consider the associated vanishing discount problem with a non-degenerate condition and prove convergence of solutions as the discount factor goes to $0$. We also discuss the uniqueness of the discounted solution. The convergence result is a selection principle for fixed points of a family of nonlinear operators.

math.OC↗
arXiv2024

Time periodic solutions of first order mean field games from the perspective of Mather theory

In this paper, the existence of non-trivial time periodic solutions of first order mean field games is proved. It is assumed that there is a non-trivial periodic orbit contained in the Mather set. The whole system is autonomous with a monotonic coupling term. Moreover, the large time convergence of solutions of first order mean field games to time periodic solutions is also considered.

math.AP↗
arXiv2023

Aubry-Mather theory for contact Hamiltonian systems III

By exploiting the contact Hamiltonian dynamics $(T^*M\times\mathbb R,Φ_t)$ around the Aubry set of contact Hamiltonian systems, we provide a relation among the Mather set, the $Φ_t$-recurrent set, the strongly static set, the Aubry set, the Mañé set and the $Φ_t$-non-wandering set. Moreover, we consider the strongly static set, as a new flow-invariant set between the Mather set and the Aubry set, in the strictly increasing case. We show that this set plays an essential role in the representation of certain minimal forward weak KAM solution and the existence of transitive orbits around the Aubry set.

math.DS↗
arXiv2023

A nonlinear semigroup approach to Hamilton-Jacobi equations--revisited

We consider the Hamilton-Jacobi equation \[{H}(x,Du)+λ(x)u=c,\quad x\in M, \] where $M$ is a connected, closed and smooth Riemannian manifold. The functions ${H}(x,p)$ and $λ(x)$ are continuous. ${H}(x,p)$ is convex, coercive with respect to $p$, and $λ(x)$ changes the signs. The first breakthrough to this model was achieved by Jin-Yan-Zhao \cite{JYZ} under the Tonelli conditions. In this paper, we consider more detailed structure of the viscosity solution set and large time behavior of the viscosity solution on the Cauchy problem.

math.AP↗
arXiv2023

A representation formula of the viscosity solution of the contact Hamilton-Jacobi equation and its applications

Assume $M$ is a closed, connected and smooth Riemannian manifold. We consider the evolutionary Hamilton-Jacobi equation \begin{equation*} \left\{ \begin{aligned} &\partial_t u(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\quad (x,t)\in M\times(0,+\infty), \\ &u(x,0)=φ(x), \end{aligned} \right. \end{equation*} where $φ\in C(M)$ and the stationary one \begin{equation*} H(x,u(x),\partial_x u(x))=0, \end{equation*} where $H(x,u,p)$ is continuous, convex and coercive in $p$, uniformly Lipschitz in $u$. By introducing a solution semigroup, we provide a representation formula of the viscosity solution of the evolutionary equation. As its applications, we obtain a necessary and sufficient condition for the existence of the viscosity solutions of the stationary equations. Moreover, we prove a new comparison theorem depending on the neighborhood of the projected Aubry set essentially, which is different from the one for the Hamilton-Jacobi equation independent of $u$.

math.AP↗
arXiv2021

Weakly coupled Hamilton-Jacobi systems without monotonicity condition: A first step

In this paper, we mainly focus on the existence of the viscosity solutions of \begin{equation*} \left\{ \begin{aligned} &H_1(x,Du_1(x),u_1(x),u_2(x))=0,\\ &H_2(x,Du_2(x),u_2(x),u_1(x))=0. \end{aligned} \right. \end{equation*} The standard assumption for the above system is called the monotonicity condition, which requires that $H_i$ is increasing in $u_i$ and decreasing in $u_j$ for each $i,j\in\{1,2\}$ and $i\neq j$. In this paper, it is assumed that $H_i$ is either increasing or decreasing in $u_i$, and may be non-monotone in $u_j$. The existence of viscosity solutions is proved when \[\chi:=\sup_{u,v,w\in\mathbb R}\bigg|\frac{\partial_{u_2} H_1(x,0,0,u)}{\partial_{u_1} H_1(x,0,v,w)}\bigg|\cdot \sup_{u,v,w\in\mathbb R}\bigg|\frac{\partial_{u_1} H_2(x,0,0,u)}{\partial_{u_2} H_2(x,0,v,w)}\bigg|<1.\] Then we consider \begin{equation*} \left\{ \begin{aligned} &h_1(x,Du_1(x))+\Lambda_1(x)(u_1(x)-u_2(x))=c,\\ &h_2(x,Du_2(x))+\Lambda_2(x)(u_2(x)-u_1(x))=\alpha(c). \end{aligned} \right. \end{equation*} It turns out that for each $c\in\mathbb R$, there is a unique constant $\alpha(c)\in\mathbb R$ such that the above system has viscosity solutions. The function $c\mapsto \alpha(c)$ is non-increasing and Lipschitz continuous. In the appendix, the large time convergence of the viscosity solution of evolutionary weakly coupled systems is proved when $\chi<1$.

math.AP↗