A Harnack inequality for solutions of elliptic-parabolic equations

We want to prove a Harnack type inequality for solutions of strongly degenerate parabolic, or elliptic-parabolic, equations. To do that, we first define a De Giorgi class of order $p=2$ that contains the solutions of evolution equations of the types $\text{uprho}(x,t)u_t+Au=0$ and $(\text{uprho}(x,t)u{)}_t+Au=0$, where $\text{uprho}>0$ almost everywhere and $A$ is a suitable elliptic operator. For functions belonging to this class we prove an inhomogeneous parabolic Harnack inequality, i.e. a Harnack inequality that takes into account the mean value of $\text{uprho}$ in different regions of $\mathrm{\Omega }\times (0,T)$. \ As a consequence, thanks to an approximation result and a delicate passage to the limit, we are able to get a Harnack inequality for solutions, and in these cases only for solutions, of strongly degenerating parabolic equations, i.e. when $\text{uprho}⩾0$. \ As a byproduct one obtains Hölder continuity for solutions of a subclass of the first equation (i.e. $\text{uprho}(x,t)u_t+Au=0$): in particular the solutions of this subclass are Hölder continuous in the interface where $\text{uprho}$ changes its sign, from positive to zero.