On the convergence of singular perturbations of Hamilton-Jacobi equations (Q609056)

Let \(A,B\) be compact metric spaces, \(f^\varepsilon,g^\varepsilon,l^\varepsilon\) (resp. \(f,g,l\)) be bounded continuous function in \(\mathbb R^n\times \mathbb R^m\times A\times B\) with values in \(\mathbb R^n,\mathbb R^m,\mathbb R\), respectively. It is assumed that there exist two positive constants \(S,s\) such that \[ \|f^\varepsilon-f\|_{\infty}+\|g^\varepsilon-g\|_{\infty}+ \|l^\varepsilon-l\|_{\infty}\leq S\varepsilon^s \] and that \(f^\varepsilon,g^\varepsilon,l^\varepsilon\) are \(\mathbb Z^m\)-periodic in the fast variable \(y\in\mathbb R^m\). The author considers the singularly perturbed problem \[ u^\varepsilon+H^\varepsilon(x,y,\nabla_x u^\varepsilon,\tfrac{1}\varepsilon\nabla_y u^\varepsilon)=0 \quad \text{in } \mathbb R^n\times \mathbb R^m, \tag{1} \] where \[ H^\varepsilon(x,y,p_x,p_y)= \min_{b\in B} \max_{a\in A}\{-p_x\cdot f^\varepsilon(x,y,a,b,)-p_y\cdot g^\varepsilon(x,y,a,b,) -l^\varepsilon(x,y,a,b)\}. \] Under a coercivity condition on the Hamiltonians \(H^\varepsilon\) and some more regularity conditions on \(f^\varepsilon,g^\varepsilon,l^\varepsilon\), equation (1) has a unique bounded solution \(u^\varepsilon\). The uniform convergence of \(u^\varepsilon(x,y)\), as \(\varepsilon\to 0\), to a Hölder continuous solution \(u(x),\) of exponent \(\alpha\in(0,1]\), to a suitable limit equation of the form \(u+\overline H(x,\nabla_x u)=0\) in \(\mathbb R^n\) is investigated. The existence of a positive constant \(N\) such that \[ \sup_{(x,y)\in\mathbb R^n\times \mathbb R^m}|u^\varepsilon(x,y)-u(x)|\leq N\varepsilon^{\alpha \min(s,\frac{1}{4-\alpha})} \] is obtained. Applications to deterministic differential games and homogenization problems are presented.