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.

[2] U. Gianazza, N. Liao and J.M. Urbano - Improved moduli of continuity for degenerate phase transitions - Preprint, (2024), 1-32, submitted
Abstract: We substantially improve in two scenarios the current state-of-the-art modulus of continuity for weak solutions to the $N$-dimensional, two-phase Stefan problem featuring a $p-$degenerate diffusion: for $p=N\geq 3$, we sharpen it to \[ \boldsymbol{\omega}(r) \approx \exp (-c| \ln r|^{\frac1N}); \] for $p\gt\max\{2,N\}$, we derive an unexpected Hölder modulus.

[3] V. Bögelein, F. Duzaar, U. Gianazza, N. Liao - Local boundedness and higher integrability for the sub-critical singular porous medium system - Preprint, (2025), 1-63, submitted
Abstract: The gradient of weak solutions to porous medium-type equations or systems possesses a higher integrability than the one assumed in the pure notion of a solution. We settle the critical and sub-critical, singular case and complete the program.