We establish the intrinsic Harnack inequality for non-negative solutions of a class of degenerate, quasilinear, parabolic equations, including equations of the p-Laplacian and porous medium type. It is shown that the classical Harnack estimate, while failing for degenerate parabolic equations, it continues to hold in a space-time geometry intrinsic to the degeneracy. The proof uses only measure-theoretical arguments, it reproduces the classical Moser theory, for non-degenerate equations, and it is novel even in that context. Hölder estimates are derived as a consequence of the Harnack inequality. The results solve a long standing problem in the theory of degenerate parabolic equations.
This work was partially supported by I.M.A.T.I.–C.N.R. (Italy).
Emmanuele DiBenedetto was supported by a NSF grant.
Dedicated to the memory of Ennio De Giorgi
"Harnack estimates for quasi-linear degenerate parabolic differential equations." Acta Math. 200 (2) 181 - 209, 2008. https://doi.org/10.1007/s11511-008-0026-3