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] S. Ciani, U. Gianazza and Z. Li - Phragmén-Lindelöf-type theorems for functions in Homogeneous De Giorgi Classess - Preprint, (2025), 1-23, submitted
Abstract: We study Phragmén-Lindelöf-type theorems for functions $u$ in homogeneous De Giorgi classes, and we show that the maximum modulus $\mu_+(r)$ of $u$ has a power-like growth of order $\alpha\in(0,1)$ when $r\to\infty$. By proper counterexamples, we show that in general we cannot expect $\alpha$ to be $1$.

[4] V. Bögelein, F. Duzaar, U. Gianazza, N. Liao and C. Scheven - Schauder estimates for parabolic $p$-Laplace systems - Preprint, (2025), 1-75, submitted
Abstract: We establish the local Hölder regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to{\mathbb R}^k$ to the non-linear system of parabolic type \begin{equation*} \partial_t u-\operatorname{div}\Big( a(x,t)\big(\mu^2+|Du|^2\big)^\frac{p-2}2Du\Big)=0 \qquad\mbox{in $E_T$}, \end{equation*} where $p>1$, $\mu\in[0,1]$, and the coefficient $a\in L^\infty(E_T)$ is bounded below by a positive constant and is Hölder continuous in the space variable $x$. As an application, we prove Hölder estimates for the gradient of weak solutions to a doubly non-linear parabolic equation in the super-critical fast diffusion regime.