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We give a new and rigorous duality relation between two central notions ofweak solutions of nonlinear PDEs: entropy and viscosity solutions. It takes theform of the {\it nonlinear dual inequality:}\begin{equation}\int |S_t u_0-S_tv_0| \varphi_0 {\rm d} x\leq \int |u_0-v_0| G_t \varphi_0 {\rm d} x, \quad\forall \varphi_0 \geq 0, \forall u_0, \forall v_0, \end{equation}where $S_t$is the entropy solution semigroup of the anisotropic degenerate parabolicequation \begin{equation*}\partial_t u+{\rm div} F(u) = {\rm div} (A(u) Du),\end{equation*}and where we look for the smallest semigroup $G_t$ satisfying\eqref{star}. This amounts to finding an optimal weighted $L^1$ contractionestimate for $S_t$. Our main result is that $G_t$ is the viscosity solutionsemigroup of the Hamilton-Jacobi-Bellman equation\begin{equation*}\partial_t\varphi=\sup\nolimits _\xi \{F'(\xi) \cdot D \varphi+ {\rm tr}(A(\xi)D^2\varphi)\}.\end{equation*} Since weigthed $L^1$ contraction results aremainly used for possibly nonintegrable $L^\infty$ solutions $u$, the naturalspaces behind this duality are $L^\infty$ for $S_t$ and $L^1$ for $G_t$. Wetherefore develop a corresponding $L^1$ theory for viscosity solutions$\varphi$. But $L^1$ itself is too large for well-posedness, and we rigorouslyidentify the weakest $L^1$ type Banach setting where we can have it -- asubspace of $L^1$ called $L^\infty_{\rm int}$. A consequence of our results isa new and optimal domain of dependence estimate for second order anisotropicdegenerate parabolic PDEs. It is given in terms of a stochastic target problemand extends in a natural way recent results for first order hyperbolic PDEs by[N. Pogodaev, {\it J. Differ. Equ.,} 2018].

Optimal stability results and nonlinear duality for $L^\infty$ entropy and $L^1$ viscosity solutions