Showing papers by "Hung V. Tran published in 2015"
TL;DR: In this article, the authors present comparison principles, Lipschitz estimates and study state constraints problems for degenerate, second-order Hamilton-Jacobi equations with respect to state constraints.
Abstract: We present comparison principles, Lipschitz estimates and study state constraints problems for degenerate, second-order Hamilton–Jacobi equations.
49 citations
TL;DR: In this paper, the authors present a proof of qualitative stochastic homogenization for a nonconvex Hamilton-Jacobi equation by introducing a family of sub-equations and controlling solutions of the original equation by the maximal subsolutions of the latter, which have deterministic limits.
Abstract: We present a proof of qualitative stochastic homogenization for a nonconvex Hamilton–Jacobi equations. The new idea is to introduce a family of “sub-equations” and to control solutions of the original equation by the maximal subsolutions of the latter, which have deterministic limits by the subadditive ergodic theorem and maximality.
42 citations
Posted Content•
TL;DR: In this paper, a class of weakly coupled Hamilton-Jacobi systems with a specific aim to perform a qualitative analysis in the spirit of weak KAM theory is studied, and the main achievement is the definition of a family of related action functionals containing the Lagrangians obtained by duality from the Hamiltonians of the system.
Abstract: We study a class of weakly coupled Hamilton-Jacobi systems with a specific aim to perform a qualitative analysis in the spirit of weak KAM theory. Our main achievement is the definition of a family of related action functionals containing the Lagrangians obtained by duality from the Hamiltonians of the system. We use them to characterize, by means of a suitable estimate, all the subsolutions of the system, and to explicitly represent some subsolutions enjoying an additional maximality property. A crucial step for our analysis is to put the problem in a suitable random frame. Only some basic knowledge of measure theory is required, and the presentation is accessible to readers without background in probability.
14 citations
TL;DR: Cagnetti et al. as mentioned in this paper applied the method to study the large-time behavior of the solution to the obstacle problem for degenerate viscous Hamilton-Jacobi equations, and established the convergence result under rather general assumptions.
Abstract: Cagnetti, Gomes, Mitake and Tran Ann Inst H Poincare Anal Non Lineaire 32(1):183–200, 2015 introduced a new idea to study the large time behavior for degenerate viscous Hamilton–Jacobi equations. In this paper, we apply the method to study the large-time behavior of the solution to the obstacle problem for degenerate viscous Hamilton–Jacobi equations. We establish the convergence result under rather general assumptions.
9 citations
TL;DR: In this article, the average behavior of interfaces moving in stationary ergodic environments with oscillatory normal velocity which changes sign is studied, and it is shown that the solutions of the oscillatory Hamilton-Jacobi equation converge in L ∞ -weak ⋆ to a linear combination of the initial datum and the solution of several initial value problems with deterministic effective Hamiltonians, determined by the properties of the random media.
5 citations
TL;DR: In this paper, the effective Hamiltonian $\bar H$ associated with the Hamiltonian $H(p,x) = H(p)+V(x) in the periodic homogenization theory is studied.
Abstract: We look at the effective Hamiltonian $\bar H$ associated with the Hamiltonian $H(p,x)=H(p)+V(x)$ in the periodic homogenization theory. Our central goal is to understand the relation between $V$ and $\bar H$. We formulate some inverse problems concerning this relation. Such type of inverse problems are in general very challenging. In the paper, we discuss several special cases in both convex and nonconvex settings.
1 citations