Homogenization of monotone systems of Hamilton-Jacobi equations

Abstract

Abstract. In this paper we study homogenization for a class of monotone systems of first-order timedependent periodic Hamilton-Jacobi equations. We characterize the Hamiltonians of the limit problem by appropriate cell problems. Hence we show the uniform convergence of the solution of the oscillating systems to the bounded uniformly continuous solution of the homogenized system.

Random Lax–Oleinik semigroups for Hamilton–Jacobi systems

“…The above stated results are, essentially, a consequence of what is proved in [7], see Appendix A for more details.…”

“…. , m}, where suitable monotonicity conditions with respect to the u j -variables are assumed on the functions G i , see [7,15,21,23,25]. In the case under investigation, the conditions assumed on the coupling matrix imply, in particular, that each function G i is non-decreasing in u i , and non-increasing in u j for j = i, for every i ∈ {1, .…”

“…This stream of research was initiated in [7], where the authors studied homogenizatioǹ a la Lions-Papanicolaou-Varadhan [26], and pursued in [8] with the proof of time convergence results for solutions of evolutive problems, under hypotheses close to [28]. Other outputs in this vein can be found in a series of works including [29,6].…”

“…When either v or w is Lipschitz continuous in [0, T ] × R N , such hypothesis can be safely removed. Indeed, with the notation used in [7], we see that either p + 2βx or p − 2βy is bounded, uniformly with respect to the parameters α, β, η, µ, since it belongs to the super differential of v j (t, ·) at x or to the subdifferential of w j (s, ·) at y. Using the estimates (3.3) and (3.4) in [7], we conclude that both p + 2βx and p − 2βy are bounded, uniformly with respect to the parameters, and the result follows without invoking condition (A.1).…”

Scalar reduction techniques for weakly coupled Hamilton–Jacobi systems

“…We focus on the critical system, obtained taking as α the minimum value for which the systems of the above family admit viscosity subsolutions. As first pointed out in [3] and [13], it is the unique system of the family for which there are viscosity solutions on the whole torus.…”