# Gradient Estimates for weak solutions of Elliptic PDE's

##### Location
Building 1, Level 3, Room 3119

#### Abstract

Let us consider an elliptic equation of second order in variational form i.e. $$\operatorname{div}(A(x) \nabla u)=\operatorname{div} f$$ in a bounded domain $\Omega \subset \mathbb{R}^n$ where the function $f$ belongs to some suitable function space. The gradient estimates problem is very important both from a theoretical and applied point of view.

In the Seminar, we exploit what is the heart of the technique to show gradient estimates allowing the function $f$ to belong to very general function spaces. Our technique is very flexible and allows us to show the existence, uniqueness, and well-posedness of the Dirichlet problem in several classes. As an example, we will give the result in $W_0^{1, p}$.

#### Brief Biography

• Born in Catania in 1963. Degree in Mathematics (University of Catania) in 1986.
• Ph.D. in 1992 in Mathematics - Partial Differential Equations under the supervision of Professor Filippo Chiarenza - University of Catania.
• Assistant professor from 1992, Associate Professor from 1998, and Full Professor from 2007 at the University of Catania.
• Chair of Master's program in Mathematics at the University of Catania.
• Visiting Professor in Italy at the University of Bologna, University of Firenze, Politecnico di Milano, Università di Napoli, Università di Salerno, Università di Urbino, Università di Trento,
• Visiting Professor at Temple University (Philadelphia PA), MSRI (Berkeley CA), Università Autonoma de Madrid, Tokyo Metropolitan University, Tokyo University of Sciences, Chuo University in Tokyo, University of Kyoto, KAUST (Saudi Arabia), Bandung Institute of Technology, University of Coimbra (Portugal).
• Research interests are about regularity problems for elliptic PDEs and boundedness properties of integral operators acting on Morrey spaces.