Sulla regolarità dell'operatore p-Laplaciano anisotropico: alla ricerca di una teoria completa della regolarità

With this note, we aim at drawing a short, coherent, and compact guide of the state-of-the-art on the theory of basic regularity, such as local boundedness, local Hölder continuity, Harnack estimates and some of their consequences, in the context of solutions to anisotropic p-Laplacean operators, elliptic and parabolic.Con questa breve nota intendiamo proporre una panoramica, breve, coerente e compatta sullo stato dell'arte della teoria della regolarità degli operatori p-Laplaciani anisotropici sia ellittici che parabolici. Tratteremo proprietà di base quali la limitatezza locale, la continuità Hölderiana, le stime di Harnack e le loro principali conseguenze

Continuity in two dimensions for a very degenerate elliptic equation

An elliptic equation div(F(Du)) = f whose ellipticity strongly degenerates for small values of Du (say, F = 0 on B(0,1)) is considered. The aim is to prove regularity for F(Du). The paper proves a continuity result in dimension two and presents some applications

A new short proof of regularity for local weak solutions for a certain class of singular parabolic equations

We shall establish the interior H\"older continuity for locally bounded weak solutions to a class of parabolic singular equations whose prototypes are \begin{equation} u_t= \nabla \cdot \bigg( |\nabla u|^{p-2} \nabla u \bigg), \quad \text{ for } \quad 1<p<2, \end{equation} and \begin{equation} u_{t}- \nabla \cdot ( u^{m-1} | \nabla u |^{p-2} \nabla u ) =0 , \quad \text{for} \quad m+p>3-\frac{p}{N}, \end{equation} via a new and simplified proof using recent techniques on expansion of positivity and $L^{1}$-Harnack estimates.Comment: 13 pages long, references 25 title