An inhomogeneous Harnack inequality for parabolic equations and applications to elliptic-parabolic and forward-backward parabolic equations

Abstract: We define a homogeneous De Giorgi class of order $p = 2$ that contains the solutions of evolution equations of the types $\rho (x,t) u_t + A u = 0$ and $(\rho (x,t) u)_t + A u = 0$, where $\rho > 0$ almost everywhere and $A$ is a suitable elliptic operator. For functions belonging to this class we prove an inhomogeneous Harnack inequality, i.e. a Harnack inequality that takes into account the mean value of $\rho$ in different regions of $\Omega \times (0,T)$. As a consequence we get a Harnack inequality for solutions, and in these case only for solutions, of elliptic-parabolic equations and of forward-backward parabolic equations. As a byproduct one obtains H\"older continuity for solutions of a subclass the first equation: in particular the solutions of this subclass are H\"older continuous in the interface where $\rho$ changes its sign.