我们拟在更一般的条件（相对于Tonelli条件中的超线性增长性，我们在强制性条件）下，结合PDE方法和变分法，研究时间周期Hamilton-Jacobi方程及弱耦合 Hamilton-Jacobi方程组相应的discounted方程的粘性解在discount因子趋于0时的收敛情形。研究这个过程中粘性解所反映的动力学性质，以及在此种条件下建立起来的测度与Tonelli条件下Mather测度的联系与区别。

We plan to study the asymptotic behavior, as the discount factor goes to 0, of the viscosity solutions of the discounted equations of time-periodic Hamilton-Jacobi equation and weakly coupled Hamitlon-Jacobi equations. We'll study it with PDE method and variational method.under more generally conditions (Compared to superlinearity condition under Tonelli frame, we'll do it under coercivity condition). We'll probe.the dynamical property of the viscosity solutions in this process and discuss the connections and differences between the Mather measure under the Tonelli conditions and the measure we'll construct under more generally conditions.