Speaker
Description
We are concerned with the Sobolev regularity of a flow $X:\,I\times I\times \Omega\to\Omega$ associated to a non-smooth vector field $b:\,I\times\Omega\to\Omega$, i.e. the solution of the Cauchy problem
$$
\begin{cases}
\partial_tX(t,s,x)&=\,b(t,X(t,s,x))\\
X(s,x)&=\,x
\end{cases}
\quad\,t,s\in I, \,x\in\Omega\,,
\tag{P}
$$
where $\Omega\subset\mathbb R^n$ is a given open domain and $I\subset\mathbb R$ is a given interval.
We are going to discuss assumptions on vector field $b$ in order that (P) is well-posed, that is, if it admits existence and uniqueness. Moreover we will focus on the Sobolev regularity of the associated flow $X$, that is, whether, for a given $p\ge\,1$, $X(t,s,\cdot)\in W_{loc}^{1,p}(\Omega_{(t,s)},\mathbb R^n)$ for given $t,s\in I$, where $\Omega_{(t,s)}$ denotes the open set of $x\in\Omega$ such that the path starting at $x$ at time $s$ can be extended until time $t$. We will review some well-known results in this topic and we will present some new results which are part of a joint work with L. Ambrosio and S. Nicolussi Golo (Jyväskylä). Eventually an application will be given to the Bernstein problem for area-minimizing intrinsic graphs in the sub-Riemannian first Heisenberg group, which is part of a joint work with S. Nicolussi Golo and Mattia Vedovato (Trento) still in progress.
