Archive for Rational Mechanics and Analysis ( IF 2.5 ) Pub Date : 2021-05-24 , DOI: 10.1007/s00205-021-01667-y Ya-Nan Wang , Jun Yan , Jianlu Zhang
For any compact connected manifold M, we consider the generalized contact Hamiltonian H(x, p, u) defined on \(T^*M\times \mathbb {R}\) which is convex in p and monotonically increasing in u. Let \(u_\varepsilon ^-:M\rightarrow \mathbb {R}\) be the viscosity solution of the parametrized contact Hamilton–Jacobi equation
$$\begin{aligned} H(x,d_x u_\varepsilon ^-(x),\varepsilon u_\varepsilon ^-(x))=c(H), \end{aligned}$$with c(H) being the Mañé Critical Value. We prove that \(u_\varepsilon ^-\) converges uniformly, as \(\varepsilon \rightarrow 0_+\), to a specific viscosity solution \(u_0^-\) of the critical equation
$$\begin{aligned} H(x,d_x u_0^-(x),0)=c(H), \end{aligned}$$which can be characterized as a minimal combination of the associated Peierls barrier functions.
中文翻译：
广义接触式Hamilton–Jacobi方程的粘滞解的收敛性
对于任何紧凑的连通歧管M，我们考虑在\（T ^ * M \ times \ mathbb {R} \）上定义的广义接触哈密顿量H（x， p， u），它在p中凸并且在u中单调增加。令\（u_ \ varepsilon ^-：M \ rightarrow \ mathbb {R} \）为参数化接触汉密尔顿–雅各比方程的粘度解
$$ \ begin {aligned} H（x，d_x u_ \ varepsilon ^-（x），\ varepsilon u_ \ varepsilon ^-（x））= c（H），\ end {aligned} $$与Ç（ħ）作为鬃毛临界值。我们证明\（u_ \ varepsilon ^-\）一致收敛为\（\ varepsilon \ rightarrow 0 _ + \）到临界方程的特定粘度解\（u_0 ^-\）
$$ \ begin {aligned} H（x，d_x u_0 ^-（x），0）= c（H），\ end {aligned} $$可以将其描述为相关的Peierls势垒函数的最小组合。
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