Atsushi Nakayasu, Tokinaga Namba
We investigate asymptotic behaviors of a metric viscosity solution of a Hamilton-Jacobi equation defined on a general metric space in Gangbo-Świȩch sense. Our results include general stability and large time behavior of the solution.
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We study the distance function to the boundary, Finsler geometry and the singular set of viscosity solutions of some Hamilton-Jacobi equations.
We present comparison principles, Lipschitz estimates and study state constraints problems for degenerate, second-order Hamilton-Jacobi equations.
A new concept of viscosity solutions, namely, the Hausdorff continuous viscosity solution for the Hamilton-Jacobi equation is defined and investigated. It is shown that the main ideas within the classical theory of continuous viscosity solutions can be extended to the wider space of Hausdorff continuous functions while also generalizing some of the existing concepts of discontinuous solutions.
We establish that a viscosity solution to a multidimensional Hamilton-Jacobi equation with a convex non-degenerate hamiltonian and Bohr almost periodic initial data decays to its infimum as time $t\to+\infty$.
We study the representation formulae for the fundamental solutions and viscosity solutions of the Hamilton-Jacobi equations of contact type. We also obtain a vanishing contact structure result for relevant Cauchy problems which can be regarded as an extension to the vanishing discount problem.
For a Hamilton-Jacobi equation defined on a network, we introduce its vanishing viscosity approximation. The elliptic equation is given on the edges and coupled with Kirchhoff-type conditions at the transition vertices. We prove that there exists exactly one solution of this elliptic approximation and mainly that, as the viscosity vanishes, it converges to the unique solution of the original problem.
We consider Hamilton--Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results are the existence and well--posedness of a viscosity solution to the Cauchy problem. We define a viscosity solution by treating the discontinuities in the coefficients analogously to ``internal boundaries''. By defining an appropriate penalization function, we prove that viscosity solutions are unique. The existence of viscosity solutions is established by showing that a sequence of front tracking approximations is compact in $L^\infty$, and that the…
We establish that a viscosity solution to a multidimensional Hamilton-Jacobi equation with Bohr almost periodic initial data remains to be spatially almost periodic and the additive subgroup generated by its spectrum does not increase in time. In the case of one space variable and a non-degenerate hamiltonian we prove the decay property of almost periodic viscosity solutions when time $t\to+\infty$. For periodic solutions the more general result is proved on unconditional asymptotic convergence of a viscosity solution to a traveling wave.
We investigate the large-time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton--Jacobi equations on the $n$-dimensional torus. We establish a convergence result to asymptotic solutions as time goes to infinity under rather restricted assumptions.
Some properties of characteristic curves in connection with viscosity solutions of Hamilton-Jacobi equations defined by Hopf-type formula are studied. We investigate the points where the Hopf-type formula $u(t,x)$ is differentiable, and the strip of the form $(0,t_0)\times \R^n$ of the domain $Ω$ where the viscosity solution $u(t,x)$ is continuously differentiable. Moreover, we present the propagation of singularity in forward of $u(t,x).$
In this paper a duality relation between the Mañé potential and Mather's action functional is derived in the context of convex and state-dependent Hamiltonians. The duality relation is used to obtain min-max representations of viscosity solutions of first order Hamilton-Jacobi equations. These min-max representations naturally suggest class\-es of subsolutions of Hamilton-Jacobi equations that arise in the theory of large deviations. The subsolutions, in turn, are good candidates for designing efficient rare-event simulation algorithms.
Sharp temporal decay estimates are established for the gradient and time derivative of solutions to a viscous Hamilton-Jacobi equation as well the associated Hamilton-Jacobi equation. Special care is given to the dependence of the estimates on the viscosity. The initial condition being only continuous and either bounded or non-negative. The main requirement on the Hamiltonians is that it grows superlinearly or sublinearly at infinity, including in particular H(r) = r^p for r non-negatif and p positif and different from 1.
Lower gradient estimates for viscosity solutions to first-order Hamilton--Jacobi equations depending on the unknown function
本文研究Hamilton量依赖于未知函数的一阶Hamilton--Jacobi方程，通过粘性解与Barron--Jensen解的等价性及Lie方程两种方法，建立了粘性解梯度的下界估计。
This paper studies Hamilton-Jacobi equations of evolution type defined in a general metric space. We give a notion of a solution through optimal principles and establish a unique existence theorem of the solution for initial value problems. We also note a relationship between the notion of a solution and another notion based on characterization of the modulus of the gradient in the sense of [13].
This paper is concerned with the qualitative properties of viscocity solutions to a class of Hamilton-Jacobi equations (HJEs) in Banach spaces. Specifically, based on the concept of $β$-derivative \cite{DGZ93b} we establish the existence, uniqueness and stability of $β$-viscosity solutions for a class of HJEs in the form $u+H(x,u,Du)=0.$ The obtained results in this paper extend ealier works in the literature, for example, \cite{CL85}, \cite{CL86} and \cite{DGZ93b}.
We consider the simplest example of a time-dependent first order Hamilton-Jacobi equation, in one space dimension and with a bounded and Lipschitz continuous Hamiltonian which only depends on the spatial derivative. We show that if the initial function has a finite number of jump discontinuities, the corresponding discontinuous viscosity solution of the corresponding Cauchy problem on the real line is unique. Uniqueness follows from a comparison theorem for semicontinuous viscosity sub- and supersolutions, using the barrier effect of spatial discontinuities of a solution. We also prove an exis…
In quantitative genetics, viscosity solutions of Hamilton-Jacobi equations appear naturally in the asymptotic limit of selection-mutation models when the population variance vanishes. They have to be solved together with an unknown function I(t) that arises as the counterpart of a non-negativity constraint on the solution at each time. Although the uniqueness of viscosity solutions is known for many variants of Hamilton-Jacobi equations, the uniqueness for this particular type of constrained problem was not resolved, except in a few particular cases. Here, we provide a general answer to the un…
We construct an explicit representation of viscosity solutions of the Cauchy problem for the Hamilton-Jacobi equation $(H,σ)$ on a given domain $Ω= (0,T)\times \R^n.$ It is known that, if the Hamiltonian $H = H(t,p)$ is not a convex (or concave) function in $p$, or $H(\cdot, p)$ may change its sign on $(0,T)$, then the Hopf-type formula does not define a viscosity solution on $Ω.$ Under some assumptions for $H(t,p)$ on the subdomains $(t_i, t_{i+1})\times \R^n\subset Ω$, we are able to arrange "partial solutions" given by the Hopf-type formula to get a viscosity solution on $Ω.$ Then we study …
Recently I. Capuzzo Dolcetta, F. Leoni and A. Porretta obtain a very surprising regularity result for fully nonlinear, superquadratic, elliptic equations by showing that viscosity subsolutions of such equations are locally Hölder continuous, and even globally if the boundary of the domain is regular enough. The aim of this paper is to provide a simplified proof of their results, together with an interpretation of the regularity phenomena, some extensions and various applications.
We present two new sharp regularity results (regularizing effect and propagation of regularity) for viscosity solutions of uniformly convex space homogeneous Hamilton-Jacobi equations. In turn, these estimates yield new intermittent stochastic regularization results for pathwise (stochastic) viscosity solutions of Hamilton-Jacobi equations with uniformly convex Hamiltonians and rough multiplicative time dependence. Finally, we use the intermittent estimates to study the long time behavior of the pathwise (stochastic) viscosity solutions of convex Hamilton-Jacobi equations.
A representation formula for viscosity solutions of nonlocal Hamilton--Jacobi equations and applications
针对速度依赖包围区域非局部量的闭合曲面几何运动，论文借助水平集方法研究非局部 Hamilton–Jacobi 方程，建立基于控制的粘性解表示公式，并用于分析增肥现象与大时间渐近性。
Remarks on the rate of convergence of the vanishing viscosity process of Hamilton-Jacobi equations
本文通过半拉普拉斯算子正则化PDE，为Hamilton-Jacobi方程半凹解的粘性消失过程建立了线性 $L^p$ 收敛速度，改进了经典二阶正则化估计，并获得了比变量加倍技术更快的收敛速度。
Vanishing discount and viscosity selection problems for mechanical Hamilton--Jacobi equations
本文研究力学Hamilton--Jacobi方程在临界、次临界和超临界情形下的消失折扣与粘性极限，明确刻画临界选择极限，并证明收敛性与Gibbs测度集中现象。
We consider an initial value problem for a Hamilton--Jacobi equation with a quadratic and degenerate Hamiltonian. Our Hamiltonian comes from the dynamics of $N$-peakon in the Camassa--Holm equation. It is given by a quadratic form with a singular positive semi-definite matrix. Such a problem does not fall into the standard theory of viscosity solutions. Also viability related results, sometimes used to deal with degenerate Hamiltonians, do not seem applicable in our case. We prove the global existence of a viscosity solution by looking at the associated optimal control problem and showing that…
This work is devoted to the studies of a Hamilton--Jacobi equation with a quadratic and degenerate Hamiltonian, which comes from the dynamics of a multipeakon in the Camassa--Holm equation. It is given by a quadratic form with a singular positive semi-definite matrix. We increase the regularity of the value function considered in our previous paper, which is known to be the viscosity solution. We prove that for a two-peakon Hamiltonian such solutions are actually $1/2$-Hölder continuous in space and time-Lipschitz continuous. The time-Lipschitz regularity is proven in any dimension $N\geq 1$. …
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 …
Three definitions of viscosity solutions for Hamilton-Jacobi equations on networks recently appeared in literature ([1,4,6]). Being motivated by various applications, they appear to be considerably different. Aim of this note is to establish their equivalence.
We consider the Cauchy problem for the Hamilton-Jacobi equation with critical dissipation, $$ \partial_t u + (-Δ)^{ 1/2} u = |\nabla u|^p, \quad x \in \mathbb R^N, t > 0, \qquad u(x,0) = u_0(x) , \quad x \in \mathbb R^N, $$ where $p > 1$ and $u_0 \in B^1_{r,1}(\mathbb R^N) \cap B^1_{\infty,1} (\mathbb R^N)$ with $r \in [1,\infty]$. We show that for sufficiently small $u_0 \in \dot B^1_{\infty,1}(\mathbb R^N)$, there exists a global-in-time mild solution. Furthermore, we prove that the solution behaves asymptotically like suitable multiplies of the Poisson kernel.
The objective of this paper is to present some results about viscosity subsolutions of the contact Hamiltonian-Jacobi equations on connected, closed manifold $M$ $$ H(x,\partial_x u,u)= 0, \quad x\in M. $$ Based on implicit variational principles introduced in [12,14], we focus on the monotonicity of the solution semigroups on viscosity subsolutions and the positive invariance of the epigraph for viscosity subsolutions. Besides, we show a similar consequence for strict viscosity subsolutions on $M$.
The existence and stability of viscosity solutions to perturbed contact Hamilton-Jacobi equations
该文研究带接触变量依赖的接触哈密顿-雅可比方程，证明局部Lyapunov渐近稳定的粘性解在小扰动下仍存在一致收敛的粘性解，且在邻域内解唯一并保持渐近稳定性。
Convergence Rates for Vanishing Viscosity Approximations of Possibly Degenerate Viscous Hamilton--Jacobi Equations
研究平坦环面上可能退化的粘性哈密顿 - 雅可比方程的消失粘性逼近的收敛速度，在哈密顿量标准假设下，先证逐点$O(\epsilon|\log\epsilon|)$收敛速度，后证测试误差时对数 Loss 可去，有平均$O(\epsilon)$速度，基于非线性伴随方法等证明。
Convergence rates for the vanishing viscosity approximation of Hamilton-Jacobi equations: the convex case
本文针对凸 Hamilton-Jacobi 方程的消失粘性逼近，利用积分方法证明了 $L^\infty$ 范数下 $\mathcal{O}(ε^β)$ 或 $\mathcal{O}(ε|\logε|)$ 的收敛速率，改进了此前 $L^p$ 范数结果并避免使用随机控制或最大值原理。
For each continuous initial data $φ(x)\in C(M,\mathbb{R})$, we obtain the asymptotic Lipschitz regularity of the viscosity solution of the following evolutionary Hamilton-Jacobi equation with convex and coercive Hamiltonians: \partial_tu(x,t)+H(x,\partial_xu(x,t))=0, u(x,0)=φ(x).
Based on a fixed point argument, we give a {\it dynamical representation} of the viscosity solution to Cauchy problem of certain weakly coupled systems of Hamilton-Jacobi equations with continuous initial datum. Using this formula, we obtain some regularity results related to the viscosity solution, including a partial extension of Lions' regularizing effect \cite{L} to the case of weakly coupled systems.
On equivalence of entropy and viscosity solutions to degenerate parabolic equations and applications
该研究证明了各向异性退化抛物-双曲方程与退化拟线性Hamilton-Jacobi方程的熵解和粘性解的等价性，并应用该结论得到周期情形下两类方程对应解的大时间行为。
Pathological viscosity solutions of Hamilton--Jacobi equations
本文在四维单位球上构造连续函数和光滑哈密顿量，使得同一函数对任意常数c均满足粘性方程，揭示粘性解中右端项不确定的病理现象。
For non convex Hamiltonians, the viscosity solution and the more geometric minimax solution of the Hamilton-Jacobi equation do not coincide in general. They are nevertheless related: we show that iterating the minimax procedure during shorter and shorter time intervals one recovers the viscosity solution.
We extend the Barles-Perthame procedure of semi-relaxed limits of viscosity solutions of Hamilton-Jacobi equations of the type f - lambda H f = h. The convergence result allows for equations on a `converging sequence of spaces' as well as Hamilton-equations written in terms of two equations in terms of operators H_\dagger and H_\dagger that serve as natural upper and lower bounds for the `true' operator H. In the process, we establish a strong relation between non-linear pseudo-resolvents and viscosity solutions of Hamilton-Jacobi equations. As a consequence we derive a convergence result for …
We establish a framework that allows to prove Gamma-converge of functionals of Lagrangian form on spaces of trajectories based on convergence of viscosity solutions of associated Hamilton-Jacobi equations. Gamma convergence follows from a: equi-coercivity, b: Gamma convergence of the projected functional at time 0, c: convergence of the Hamiltonians that appear as Legendre transform of the Lagrangian in the path-space functional.
A convergence theorem for Crandall-Lions viscosity solutions to path-dependent Hamilton-Jacobi-Bellman PDEs
本文针对路径依赖Hamilton-Jacobi-Bellman偏微分方程，建立了Crandall-Lions粘性解的收敛定理，其证明依赖于新的动态次线性期望收敛定理和粘性解的值函数随机表示。
Uniqueness of semi-concave weak solutions for Hamilton-Jacobi equations
本文受spin glasses问题启发，证明了具有严格凸非线性项和足够正则初始条件的Hamilton-Jacobi PDE的半凹弱解即为粘性解，从而确立了其唯一性。
Viscosity Solutions for Hamilton-Jacobi equations on Separable Banach Spaces
该研究构建了可分巴拿赫空间最优控制问题的粘性解处理框架，扩展了B-粘性解概念，证明严格比较原理并将最优控制问题的值函数表征为无限维HJB方程的唯一粘性解。
Optimal rate of convergence in the vanishing viscosity for uniformly convex Hamilton-Jacobi equations
本文针对一致凸 Hamilton–Jacobi 方程的消失粘性极限，结合上卷积正则化与伴随线性化流的熵估计，证明半凹 Lipschitz 终端条件下最优收敛速率为 O(εlog ε)。
Rate of convergence of the vanishing viscosity method for Hamilton-Jacobi equations with Neumann boundary conditions
本文研究了带有Neumann边界条件的Hamilton-Jacobi方程的Vanishing Viscosity方法收敛速率，针对不同Hamiltonian类型给出了不同收敛速度的理论分析。
Restriction of viscosity solutions of the Hamilton-Jacobi equation to a submanifold M of \mathbb{R}^{d}
该研究界定 HJ 方程粘性解对子流形的限制含义，并在 Hamilton 向量场保持 M×R^d 不变的条件下给出相应处理。
Suppose $M$ is a closed Riemannian manifold. For a $C^2$ generic (in the sense of Mañé) Tonelli Hamiltonian $H: T^*M\rightarrow\mathbb{R}$, the minimal viscosity solution $u_λ^-:M\rightarrow \mathbb{R}$ of the negative discounted equation \[-λu+H(x,d_xu)=c(H),\quad x\in M,\ λ>0 \] with the Mañé's critical value $c(H)$ converges to a uniquely established viscosity solution $u_0^-$ of the critical Hamilton-Jacobi equation \[ H(x,d_x u)=c(H),\quad x\in M \] as $λ\rightarrow 0_+$. We also propose a dynamical interpretation of $u_0^-$.
Remarks on the vanishing viscosity process of state-constraint Hamilton-Jacobi equations
本文研究了次二次状态约束 Hamilton-Jacobi 方程解在消失粘性过程中的收敛速率，通过分析边界附近的爆破行为与解的半凹性，证明了在不同数据条件下内部及单侧收敛速率的改进结果。
On Viscosity Solutions of Hamilton-Jacobi Equations in the Wasserstein space and the Vanishing Viscosity Limit
本文研究了Wasserstein空间中Hamilton-Jacobi方程的粘性解及其粘性消失极限，提出了统一框架并证明了最优收敛速率。
In this paper we study the fully nonlinear stochastic Hamilton-Jacobi-Bellman (HJB) equation for the optimal stochastic control problem of stochastic differential equations with random coefficients. The notion of viscosity solution is introduced, and we prove that the value function of the optimal stochastic control problem is the maximal viscosity solution of the associated stochastic HJB equation. For the superparabolic cases when the diffusion coefficients are deterministic functions of time, states and controls, the uniqueness is addressed as well.
We prove the uniqueness of the viscosity solution to the Hamilton-Jacobi equation associated with a Bolza problem of the Calculus of Variations, assuming that the Lagrangian is autonomous, continuous, superlinear, and satisfies the usual convexity hypothesis. Under the same assumptions we prove also the uniqueness, in a class of lower semicontinuous functions, of a slightly different notion of solution, where classical derivatives are replaced only by subdifferentials. These results follow from a new comparison theorem for lower semicontinuous viscosity supersolutions of the Hamilton-Jacobi eq…
We discuss various kinds of representation formulas for the viscosity solutions of the contact type Hamilton-Jacobi equations by using the Herglotz' variational principle.
Time periodic and almost periodic viscosity solutions of contact Hamilton-Jacobi equations on $\mathbb{T}^n$
本文研究了环面上发展型接触 Hamilton-Jacobi 方程的时间周期粘性解问题，证明了非平凡 $T$-周期解的存在性条件，并探讨了 Bohr 概周期粘性解的存在性。
We prove the comparison principle for viscosity sub and super solutions of degenerate nonlocal operators with general nonlocal gradient nonlinearities. The proofs apply to purely Hamilton-Jacobi equations of order $0<s<1$.
It is well-known that solutions to the basic problem in the calculus of variations may fail to be Lipschitz continuous when the Lagrangian depends on t. Similarly, for viscosity solutions to time-dependent Hamilton-Jacobi equations one cannot expect Lipschitz bounds to hold uniformly with respect to the regularity of coefficients. This phenomenon raises the question whether such solutions satisfy uniform estimates in some weaker norm. We will show that this is the case for a suitable Hölder norm, obtaining uniform estimates in (x,t) for solutions to first and second order Hamilton-Jacobi equat…
Convergence rates for the vanishing viscosity approximation of fully nonlinear, non-convex, second-order Hamilton-Jacobi equations
本文针对全非线性非凸退化 Hamilton-Jacobi 方程，利用粘性解上/下卷积正则化性质与比较原理，在解及 Hamiltonian 无可微性假设下，建立了显式的消失粘性逼近收敛速度估计。
We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi equations with large drift terms, where the drift terms are given by the Hamiltonian vector fields of Hamiltonian $H$. This is an attempt to understand the averaging effect for fully nonlinear degenerate elliptic equations. In this work, we restrict ourselves to the case of Hamilton-Jacobi equations. The second author has already established averaging results for Hamilton-Jacobi equations with convex Hamiltonians ($G$ below) under the classical formulation of the Dirichlet condition. Here we treat the Di…
On the (non) existence of viscosity solutions of multi-time Hamilton-Jacobi equations
本文证明了在非凸设定下，即使 Hamiltonians 处于对合状态，多时间 Hamilton-Jacobi 方程通常也无法在粘性意义下求解。
We provide an example of a Hamilton-Jacobi equation in which stochastic homogenization does not occur. The Hamiltonian involved in this example satisfies the standard assumptions of the literature, except that it is not convex.
Hamilton-Jacobi equations in metric spaces
本文为度量空间中Hamilton-Jacobi方程的讲义，概述了其适定性，重点介绍程函方程的度量粘性解概念，并补充了度量空间基础与欧氏粘性解理论。
We study homogenization for a class of stationnary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, supplemented at the origin with an effective Dirichlet condition that keeps track of the perturbation. Various comments and extensions are discussed.
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