| Abstract: |
| In this talk, we discuss non-diagonal vectorial quasilinear degenerate elliptic systems defined on a bounded open set $\Omega \subset mathbb{R}^n$, with $n > 2$, of the form
\begin{equation} \label{eq_divergenza}
\begin{cases}
-
\text{div} (a(x,u(x))Du(x))= f(x) ,& x \in \Omega \
u=0, & x \in \partial \Omega
\end{cases}
\end{equation}
where $u, f : \Omega \to \mathbb{R}^N$ and
$a: \Omega \times \mathbb{R}^N \to \mathbb{R}^{N^2 n^2}$ is a bounded Carath\`{e}odory function.
Unlike the scalar case, in the vectorial setting $(N \ge 2)$ one cannot generally expect boundedness or high integrability of solutions, even for regular data. We will show how, by imposing conditions on the support of the off-diagonal coefficients, it is possible to guarantee existence of solutions whose regularity depends on the datum f.
In particular, we will analyze the case where f belongs to an appropriate Marcinkiewicz space, highlighting regularity properties for the systems under consideration. |
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