Alternative forms of the Harnack inequality for non-negative solutions to certain degenerate and singular parabolic equations

Non-negative solutions to quasi-linear, degenerate or singular parabolic partial differential equations, of p-Laplacian type for p > 2N/N+1, satisfy Harnack-type estimates in some intrinsic geometry. Some equivalent alternative forms of these Harnack estimates are established, where the supremum and the infimum of the solutions play symmetric roles, within a properly redefined intrinsic geometry. Such equivalent forms hold for the non-degenerate case p = 2 following the classical work of Moser , and are shown to hold in the intrinsic geometry of these degenerate and/or parabolic p.d.e.'s. Some new forms of such an estimate are also established for 1 < p > 2.