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.
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