Related papers: Weak KAM theory for general Hamilton-Jacobi equati…

We study the Cauchy problem of a Hamilton-Jacobi equation with the spatial variable in a closed convex cone. A monotonicity assumption on the nonlinearity allows us to prescribe no condition on the boundary of the cone. We show the…

We study the periodic homogenization of the viscous Hamilton--Jacobi equation \[ u_t^\varepsilon + \frac{1}{2}|Du^\varepsilon|^2 + V\!\left(\frac{x}{\varepsilon}\right) = \frac{\varepsilon}{2}\Delta u^\varepsilon \qquad \text{in }…

We consider the Hamilton-Jacobi equation \[{H}(x,Du)+\lambda(x)u=c,\quad x\in M, \] where $M$ is a connected, closed and smooth Riemannian manifold. The functions ${H}(x,p)$ and $\lambda(x)$ are continuous. ${H}(x,p)$ is convex, coercive…

We develop an approach for the analysis of fundamental solutions to Hamilton-Jacobi equations of contact type based on a generalized variational principle proposed by Gustav Herglotz. We also give a quantitative Lipschitz estimate on the…

We consider a sequence of finite irreducible Markov chains with exponentially small transition rates: the transition graph is a fixed, finite, strongly connected directed graph; the transition rates decay exponentially on a paramenter N…

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…

This paper concerns with the time periodic viscosity solution problem for a class of evolutionary contact Hamilton-Jacobi equations with time independent Hamiltonians on the torus $\mathbb{T}^n$. Under certain suitable assumptions we show…

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…

We propose notions of minimax and viscosity solutions for a class of fully nonlinear path-dependent PDEs with nonlinear, monotone, and coercive operators on Hilbert space. Our main result is well-posedness (existence, uniqueness, and…

Suppose that $H(x,u,p)$ is strictly decreasing in $u$ and satisfies Tonelli conditions in $p$. We show that each viscosity solution of $H(x,u,u_x)=0$ can be reached by many viscosity solutions of $$ w_t+H(x,w,w_x)=0, $$ in a finite time.

In this article, we are interested in the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi Equations. In the superquadratic case, the third author has proved that these solutions can have…

In the paper we prove the convergence of viscosity solutions $u_{\lambda}$ as $\lambda\rightarrow0_+$ for the parametrized degenerate viscous Hamilton-Jacobi equation \[ H(x,d_x u, \lambda u)=\alpha(x)\Delta u,\quad \alpha(x)\geq 0,\quad…

This paper presents an implicit solution formula for the Hamilton-Jacobi partial differential equation (HJ PDE). The formula is derived using the method of characteristics and is shown to coincide with the Hopf and Lax formulas in the case…

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…

We consider the Cauchy problem for a strictly hyperbolic, $n\times n$ system in one space dimension: $u_t+A(u)u_x=0$, assuming that the initial data has small total variation. We show that the solutions of the viscous approximations…

In this paper, we establish the convergence of solutions to the viscous Hamilton-Jacobi equation (with a Tonelli Hamiltonian): \[ \lambda u +H(x, du)=\varepsilon(\lambda)\Delta u,\quad \lambda>0 \] as $\lambda\rightarrow 0_+$, once the…

We study minimax (generalized) solutions of a Cauchy problem for a (first-order) path-dependent Hamilton--Jacobi equation with co-invariant derivatives under a right-end boundary condition. Under assumptions on the Hamiltonian that are more…

In this paper, we develop a universal, conceptually simple and systematic method to prove well-posedness to Cauchy problems for weak solutions of parabolic equations with non-smooth, time-dependent, elliptic part having a variational…

We consider a Cauchy problem for a Hamilton--Jacobi equation with coinvariant derivatives of an order $\alpha \in (0, 1)$. Such problems arise naturally in optimal control problems for dynamical systems which evolution is described by…

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…