CNA Seminar

Paolo Marcellini
Universita' di Firenze
Title: Nonlinear elliptic systems with general growth

Abstract: We give some recent existence and interior regularity results for elliptic partial differential equations in divergence form of the type

\begin{equation*} \sum_{i=1}^{n}\frac{\partial }{\partial x_{i}}a^{i}\left( x,u\left( x\right) ,Du\left( x\right) \right) =b\left( x,u\left( x\right) ,Du\left( x\right) \right) \,, \end{equation*}

for $x\in \Omega $, open set of $\mathbb{R}^{n}$, $n\geq 2$, related to a vector field $a^{i}\left( x,s,\xi \right) $ locally Lipschitz continuous in $% \Omega \times \mathbb{R}\times \mathbb{R}^{n}$.

More generally we consider elliptic systems of $m$ partial differential equations in divergence form of the type \begin{equation*} \sum_{i=1}^{n}\frac{\partial }{\partial x_{i}}a_{\alpha }^{i}\left( x,u\left( x\right) ,Du\left( x\right) \right) =b_{\alpha }\left( x,u\left( x\right) ,Du\left( x\right) \right) ,\;\;\;\;\alpha =1,2,\ldots ,m\,, \end{equation*}% for maps $u:\Omega \subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$. Here the vector field $\left( a_{\alpha }^{i}\left( x,s,\xi \right) \right) $ assumes values in the set of $m\times n$ matrices and it satisfies some \emph{general growth conditions} with respect to the gradient variable $\xi \in \mathbb{R}^{m\times n}$.

Recording (avi): http://mm.math.cmu.edu/recordings/cna/CNA-PaoloMarcellini-Sep-18-2012.avi
Recording (mp4): http://mm.math.cmu.edu/recordings/cna/CNA-PaoloMarcellini-Sep-18-2012.mp4
Pdf File: Marcellini_Paolo.pdf
Date: Tuesday, September 18, 2012
Time: 1:30 pm
Location: Wean Hall 7218
Submitted by:  David Kinderlehrer