Related papers: Stochastic homogenization for variational solution…

We consider the following evolutionary Hamilton-Jacobi equation with initial condition: \begin{equation*} \begin{cases} \partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\ u(x,0)=\phi(x), \end{cases} \end{equation*} where $\phi(x)\in…

We are interested in the averaged behavior of interfaces moving in stationary ergodic environments, with oscillatory normal velocity which changes sign. This problem can be reformulated, using level sets, as the homogenization of a…

We study the stochastic homogenization for a Cauchy problem for a first-order Hamilton-Jacobi equation whose operator is not coercive w.r.t. the gradient variable. We look at Hamiltonians like $H(x,\sigma(x)p,\omega)$ where $\sigma(x)$ is a…

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…

We study random homogenization of second-order, degenerate and quasilinear Hamilton-Jacobi equations which are positively homogeneous in the gradient. Included are the equations of forced mean curvature motion and others describing…

This paper establishes a complete homogenization theory for the one-dimensional parabolic equation with long-range correlated random potential: \[ \partial_t u_\varepsilon(t,x) = \frac{1}{2} \partial_{xx} u_\varepsilon(t,x) +…

We present stochastic homogenization results for viscous Hamilton-Jacobi equations using a new argument which is based only on the subadditive structure of maximal subsolutions (solutions of the "metric problem"). This permits us to give…

We estimate the variance of the value function for a random optimal control problem. The value function is the solution $w^\epsilon$ of a Hamilton-Jacobi equation with random Hamiltonian $H(p,x,\omega) = K(p) - V(x/\epsilon,\omega)$ in…

In this paper, we consider the following Hamilton-Jacobi equation with initial condition: \begin{equation*} \begin{cases} \partial_tu(x,t)+H(x,t,u(x,t),\partial_xu(x,t))=0, u(x,0)=\phi(x). \end{cases} \end{equation*} Under some assumptions…

In the paper we consider the solution of an advection equation with rapidly changing coefficients $\partial_t u_\eps+(1/\eps)V(t\eps^{-2},x/{\eps})\cdot\nabla_x u_\eps=0$ for $t<T$ and $u_\eps(T,x)=u_0(x)$, $x\in\bbR^d$. Here $\eps>0$ is…

We consider the following evolutionary Hamilton-Jacobi equation with initial condition: \begin{equation*} \begin{cases} \partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\ u(x,0)=\phi(x), \end{cases} \end{equation*} where $\phi(x)\in…

We consider the specified stochastic homogenization of first order evolutive Hamilton-Jacobi equations on a very simple junction, i.e the real line with a junction at the origin. Far from the origin, we assume that the considered…

We prove homogenization for degenerate viscous Hamilton-Jacobi equations in dimension one in stationary ergodic environments with a quasiconvex and superlinear Hamiltonian of fairly general type. We furthermore show that the effective…

We prove a stochastic homogenization result for a class of \emph{nonlinear} and \emph{nonlocal} variational problems in domains with many small randomly distributed (bilateral) obstacles. Our model case is a Dirichlet problem for the…

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.

Let $u^\varepsilon$ and $u$ be viscosity solutions of the oscillatory Hamilton-Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of…

We study the optimal convergence rate for homogenization problem of convex Hamilton-Jacobi equations when the Hamitonian is periodic with respect to spatial and time variables, and notably time-dependent. We prove a result similar to that…

We consider Hamilton-Jacobi equations in one space dimension with Hamiltonians of the form $H(p,x,\omega) = G(p) + \beta V(x,\omega)$, where $V(\cdot,\omega)$ is a stationary and ergodic potential of unit amplitude. The homogenization of…

We consider the stochastic heat equation $\partial_{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda)u$, with a smooth space-time stationary Gaussian random field $V(s,y)$, in dimensions $d\geq 3$, with an initial condition…

Consider a Hamiltonian system of type \[ -\Delta u=H_{v}(u,v),\ -\Delta v=H_{u}(u,v) \ \ \text{ in } \Omega, \qquad u,v=0 \text{ on } \partial \Omega \] where $H$ is a power-type nonlinearity, for instance $H(u,v)= |u|^p/p+|v|^q/q$, having…