Showing papers by "Hung V. Tran published in 2018"

Min–max formulas and other properties of certain classes of nonconvex effective Hamiltonians

Abstract: This paper is the first attempt to systematically study properties of the effective Hamiltonian $$\overline{H}$$ arising in the periodic homogenization of some coercive but nonconvex Hamilton–Jacobi equations. Firstly, we introduce a new and robust decomposition method to obtain min–max formulas for a class of nonconvex $$\overline{H}$$ . Secondly, we analytically and numerically investigate other related interesting phenomena, such as “quasi-convexification” and breakdown of symmetry, of $$\overline{H}$$ from other typical nonconvex Hamiltonians. Finally, in the appendix, we show that our new method and those a priori formulas from the periodic setting can be used to obtain stochastic homogenization for the same class of nonconvex Hamilton–Jacobi equations. Some conjectures and problems are also proposed.

The selection problem for discounted Hamilton–Jacobi equations: some non-convex cases

Abstract: Here, we study the selection problem for the vanishing discount approximation of non-convex, first-order Hamilton–Jacobi equations. While the selection problem is well understood for convex Hamiltonians, the selection problem for non-convex Hamiltonians has thus far not been studied. We begin our study by examining a generalized discounted Hamilton–Jacobi equation. Next, using an exponential transformation, we apply our methods to strictly quasi-convex and to some non-convex Hamilton–Jacobi equations. Finally, we examine a non-convex Hamiltonian with flat parts to which our results do not directly apply. In this case, we establish the convergence by a direct approach.

Large time average of reachable sets and Applications to Homogenization of interfaces moving with oscillatory spatio-temporal velocity

Abstract: We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.

On uniqueness sets of additive eigenvalue problems and applications

Abstract: In this paper, we provide a simple way to find uniqueness sets for additive eigenvalue problems of first and second order Hamilton--Jacobi equations by using a PDE approach. An application in finding the limiting profiles for large time behaviors of first order Hamilton--Jacobi equations is also obtained.

On obstacle problem for mean curvature flow with driving force

Abstract: In this paper, we study an obstacle problem associated with the mean curvature flow with constant driving force. Our first main result concerns interior and boundary regularity of the solution. We then study in details the large time behavior of the solution and obtain the convergence result. In particular, we give full characterization of the limiting profiles in the radially symmetric setting.

Rate of convergence in periodic homogenization of Hamilton-Jacobi equations: the convex setting

Abstract: We study the rate of convergence of $u^\epsilon$, as $\epsilon \to 0+$, to $u$ in periodic homogenization of Hamilton-Jacobi equations. Here, $u^\epsilon$ and $u$ are viscosity solutions to the oscillatory Hamilton-Jacobi equation and its effective equation \begin{equation*} {\rm (C)_\epsilon} \qquad \begin{cases} u_t^\epsilon+H\left(\frac{x}{\epsilon},Du^\epsilon\right)=0 \qquad &\text{in} \ \mathbb{R}^n \times (0,\infty), u^\epsilon(x,0)=g(x) \qquad &\text{on} \ \mathbb{R}^n, \end{cases} \end{equation*} and \begin{equation*} {\rm (C)} \qquad \begin{cases} u_t+\overline{H}\left(Du\right)=0 \qquad &\text{in} \ \mathbb{R}^n \times (0,\infty), u(x,0)=g(x) \qquad &\text{on} \ \mathbb{R}^n, \end{cases} \end{equation*} respectively. We assume that the Hamiltonian $H=H(y,p)$ is coercive and convex in the $p$ variable and is $\mathbb{Z}^n$-periodic in the $y$ variable, and the initial data $g$ is bounded and Lipschitz continuous.

On uniqueness sets of additive eigenvalue problems and applications

Abstract: In this paper, we provide a simple way to find uniqueness sets for additive eigenvalue problems of first and second order Hamilton--Jacobi equations by using a PDE approach. An application in finding the limiting profiles for large time behaviors of first order Hamilton--Jacobi equations is also obtained.