Ugo Gianazza's Latest Preprints

[1] V. Bögelein, F. Duzaar, U. Gianazza, N. Liao and C. Scheven - Hölder Continuity of the Gradient of Solutions to Doubly Non-Linear Parabolic Equations - Preprint, (2023), 1-142, submitted
Abstract: This paper is devoted to studying the local behavior of non-negative weak solutions to the doubly non-linear parabolic equation \[ \partial_t u^q - \operatorname{div}\big(|D u|^{p-2}D u\big) = 0 \] in a space-time cylinder. Hölder estimates are established for the gradient of its weak solutions in the super-critical fast diffusion regime $0\lt p-1\lt q\lt\frac{N(p-1)}{(N-p)_+}$. Moreover, decay estimates are obtained for weak solutions and their gradient in the vicinity of possible extinction time. Two main components towards these regularity estimates are a time-insensitive Harnack inequality that is particular about this regime, and Schauder estimates for the parabolic $p$-Laplace equation.