Non-negative solutions to quasi-linear, degenerate or singular parabolic partial differential equations, of p-Laplacian type for p > 2_N_/(N+1), satisfy Harnack-type estimates in some intrinsic geometry ([2, 3]). 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 ([5, 6]), 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.
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Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri, Alternative Forms of the Harnack Inequality for Non-Negative Solutions to Certain Degenerate and Singular Parabolic Equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 20 (2009), no. 4, pp. 369–377DOI 10.4171/RLM/552