Compare L1 and L2 solutions

For $J_1$, the derivative in the positive region is $w-3+1$, which gives $w^*=2$; the subgradient condition confirms the global minimum. For $J_2$, $J_2'(w)=2w-3$, giving $w^*=\frac{3}{2}$. Both objectives are convex. $L_1$ soft-thresholding can produce exact zeros; $L_2$ gives smooth proportional shrinkage and usually does not produce exact zeros.