We revisit stability results for two central notions of weak solutions for nonlinear PDEs: entropy and viscosity solutions originally introduced for scalar conservation laws and Hamilton-Jacobi equations. Here, we consider two second order model equations, the Hamilton-Jacobi-Bellman (HJB) equation \begin{equation*}\partial_t \varphi=\sup_\xi \{b(\xi) \cdot D \varphi+\mathrm{tr}(a(\xi) D^2\varphi)\},\end{equation*}and the anisotropic degenerate parabolic equation\begin{equation*}\partial_t u+\mathrm{div} F(u)=\mathrm{div} (A(u) D u).\end{equation*}The viscosity solutions of the first equation and the entropy solutions of the second satisfy contraction principles in $L^\infty$ and $L^1$ respectively. Our aim is to get similar results for viscosity solutions in $L^1$ and entropy solutions in $L^\infty$. For the first equation, we identify the smallest Banach topology which is stronger than $L^1$ for which we have stability. We then construct a norm such that a quasicontraction principle holds. For the second equation, we propose a new weighted $L^1$ contraction principle allowing for pure $L^\infty$ solutions. Our main contribution is to show that the solutions of the HJB equation can be used as weights and that this choice is optimal. Interestingly, this reveals a new type of duality between entropy and viscosity solutions.