Related papers: On almost periodic viscosity solutions to Hamilton…

We consider the Cauchy problem for the Hamilton-Jacobi equation with critical dissipation, $$ \partial_t u + (-\Delta)^{ 1/2} u = |\nabla u|^p, \quad x \in \mathbb R^N, t > 0, \qquad u(x,0) = u_0(x) , \quad x \in \mathbb R^N, $$ where $p >…

We obtain new quantitative estimates of the vanishing viscosity approximation for time-dependent, degenerate, Hamilton-Jacobi equations that are neither concave nor convex in the gradient and Hessian entries of the form $\partial_t…

We consider the stationary Hamilton-Jacobi equation where the dynamics can vanish at some points, the cost function is strictly positive and is allowed to be discontinuous. More precisely, we consider special class of discontinuities for…

We give a new perspective on the existence of viscosity solutions for a stationary and a time-dependent first-order Hamilton-Jacobi equation. Following recent comparison principles, we work in a framework in which we consider a subsolution…

It is well known that when the nonlinearity is convex, the Hamilton-Jacobi PDE admits a unique semi-convex weak solution, which is the viscosity solution. In this paper, motivated by problems arising from spin glasses, we show that if the…

We study the optimal rate of convergence in periodic homogenization of the viscous Hamilton-Jacobi equation $u^\varepsilon_t + H(\frac{x}{\varepsilon},Du^\varepsilon) = \varepsilon \Delta u^\varepsilon$ in $\mathbb R^n\times (0,\infty)$…

This paper is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi (HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity…

Let $H$ be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton-Jacobi equation $U_{t}+H(U_x)=0$ and signed Radon measure valued entropy solutions of the conservation law…

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…

The Hamilton-Jacobi equation on metric spaces has been studied by several authors; following the approach of Gangbo and Swiech, we show that the final value problem for the Hamilton-Jacobi equation has a unique solution even if we add a…

The convergence to non-diffusive self-similar solutions is investigated for non-negative solutions to the Cauchy problem $\partial_t u = \Delta_p u + |\nabla u|^q$ when the initial data converge to zero at infinity. Sufficient conditions on…

This paper deals with the periodic homogenization of nonlocal parabolic Hamilton-Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal…

We study the long-time asymptotic behavior of solutions u of the Hamilton-Jacobi equation u_t(x,t)+H(x,Du(x,t))=0 in \Omega \times (0,\infty), where \Omega is a bounded open subset of R^n, with Hamiltonian H=H(x,p) being convex and coercive…

We show strong uniform convergence of monotone P1 finite element methods to the viscosity solution of isotropic parabolic Hamilton-Jacobi-Bellman equations with mixed boundary conditions on unstructured meshes and for possibly degenerate…

In this paper, we investigate the convergence rate in the vanishing viscosity limit for solutions to superquadratic Hamilton--Jacobi equations with state constraints. For every $p>2$, we establish the rate of convergence for nonnegative…

The main purpose of this paper is to study the global propagation of singularities of viscosity solution to discounted Hamilton-Jacobi equation \begin{equation}\label{eq:discount 1}\tag{HJ$_\lambda$} \lambda v(x)+H( x, Dv(x) )=0 , \quad…

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…

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…

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.

We show in this article in what sense viscosity solutions of the Hamilton-Jacobi equation can be restricted to a submanifold M of \mathbb{R}^{d}. We treat in this article the case of M\times\mathbb{R}^{d} being invariant by the Hamiltonian…