Abstract
We will review regularity results for linear and quasilinear uniformly elliptic equations. Focus is on the minimal assumptions we need to obtain a given degree of smoothness for generalized solutions of a given elliptic equation. Our main linear equation is
$\mathrm{div} A (x, u, \nabla u) + B (x, u, \nabla u) = 0$,
where
$ |A(x,u,\xi)| \leq a |\xi|^{p-1} + b |u|^{p-1} + e$,
$ |B(x,u,\xi)| \leq c |\xi|^{p-1} + d |u|^{p-1} + g $,
$ \xi \! \cdot \! A(x,u,\xi) \geq |\xi|^{p} - d |u|^{p} - g $,
$a$ is a real constant and $b,c,d,e,g$ are given function.
Brief Biography
Prof. Giuseppe Di Fazio earned a degree in Mathematics from the University of Catania in 1986, and completed his PhD in 1992 in Mathematics, specializing in Partial Differential Equations, under the supervision of Professor Filippo Chiarenza at the University of Catania. He has been with the University of Catania since 1992, starting as an Assistant Professor, becoming an Associate Professor in 1998, and achieving the rank of Full Professor in 2007. He is currently the Chair of the Master’s program in Mathematics at the university. His research focuses on regularity problems for elliptic partial differential equations and the boundedness properties of integral operators on Morrey spaces. He is the co-author of two books:
* Morrey Spaces: Introduction and Applications to Integral Operators and PDEs, Volume I (2020) by Yoshihiro Sawano, Giuseppe Di Fazio, and Denny Ivanal Hakim (ISBN: 9781498765510) - published by Taylor & Francis.
* Morrey Spaces: Introduction and Applications to Integral Operators and PDEs, Volume II (2020) by Yoshihiro Sawano, Giuseppe Di Fazio, and Denny Ivanal Hakim (ISBN: 9780367459154) - published by Taylor & Francis.
