# Alternative Forms of the Harnack Inequality for Non-Negative Solutions to Certain Degenerate and Singular Parabolic Equations

### Emmanuele DiBenedetto

Vanderbilt University, Nashville, United States### Ugo Gianazza

Università di Pavia, Italy### Vincenzo Vespri

Universita di Firenze, Italy

## Abstract

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.

## Cite this article

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. 369–377

DOI 10.4171/RLM/552