Viscosity Solutions for Weakly Coupled Systems of Hamilton-Jacobi Equations

Abstract

We obtain comparison results for viscosity solutions of weakly coupled systems of Hamilton‐Jacobi equations. Perron's method is extended to give existence proofs for systems. A stronger definition of sub‐ and supersolutions is introduced, leading to existence results for continuous solutions. The needed sub‐ and supersolutions are constructed for continuous boundary data.

Homogenization of monotone systems of Hamilton-Jacobi equations

“…The u 0,i 's are bounded uniformly continuous (BU C in short). The monotonicity condition, see (2.3), we assume for the system is a standard assumption to obtain a comparison principle for (1.1) (see [10,[12][13][14]). The main result of the paper, see Theorem 5.2, is the convergence of u ε , as ε → 0, to a BU C function u = (u 1 , .…”

Twisted Lax–Oleinik formulas and weakly coupled systems of Hamilton–Jacobi equations

“…First, a comparison principle is established (see Proposition 2.6). This uses in an essential manner the sign properties of the coupling matrix B (they imply the system fall in a more general class of coupled systems, see [15]). This comparison principle implies uniqueness and existence follows from Perron's method (properties of B give that a supremum of subsolutions is a subsolution).…”

Some results on the large-time behavior of weakly coupled systems of first-order Hamilton–Jacobi equations

“…So we can choose a sequence t n → +∞ such that (w(·, t n + ·)) n converges uniformly to some function v ∈ W 1,∞ (T N × [0, ∞)) m . By the stability result ( [1,2,7]), v is still a viscosity solution of (3.11). We state the key estimates on the functions P η [v i ]'s.…”