A Harnack inequality for solutions of elliptic-parabolic equations

Abstract

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 $\uprho (x,t) u_t + A u = 0$ and $(\uprho (x,t) u)_t + A u = 0$, where $\uprho > 0$ 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 $\uprho$ in different regions of $\Omega \times (0,T)$. \\ 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 $\uprho \geqslant 0$. \\ As a byproduct one obtains Hölder continuity for solutions of a subclass of the first equation (i.e. $\uprho (x,t) u_t + A u = 0$): in particular the solutions of this subclass are Hölder continuous in the interface where $\uprho$ changes its sign, from positive to zero.