CVGMT Seminarshttp://cvgmt.sns.it/seminars/en-usSat, 21 Oct 2017 16:10:54 -0000Liouville-type problems on compact surfaces: a variational approachhttp://cvgmt.sns.it/seminar/599/2017-10-25: <a href="/person/1820/">A. Jevnikar</a>.
<p>A class of Liouville equations and systems on compact surfaces is considered: we focus on the Toda
system which is motivated in mathematical physics by the study of models in non-abelian
Chern-Simons theory and in geometry in the description of holomorphic curves. We discuss its
variational aspects which yield existence results.
</p>
http://cvgmt.sns.it/seminar/599/The saga of a fish: from a survival guide to closing lemmas for dynamical systemshttp://cvgmt.sns.it/seminar/600/2017-10-25: <a href="/person/26/">E. Stepanov</a>.
<p>In a recent paper by D. Burago, S. Ivanov and A. Novikov, "A survival guide for feeble fish", it has been shown that a fish with limited velocity capabilities can reach any point in the (possibly unbounded) ocean provided that the fluid velocity field is incompressible, bounded and has vanishing mean drift. This brilliant result substantially extends the celebrated point-to-point controllability theorems (in particular, the one by H. Sussmann and V. Jurdjevic), though being in a sense non constructive. We will give a fish the constructive (or "almost") recipe of how to survive in a turbulent ocean, and show how this is connected to closing lemmas for dynamical systems, in particular to C. Pugh's closing lemma, proving an extension of the latter. Joint work with Sergey Kryzhevich (St. Petersburg and Nova Gorica). </p>
http://cvgmt.sns.it/seminar/600/Monotonicity and symmetry of solutions with uniform limits for some elliptic systemshttp://cvgmt.sns.it/seminar/598/2017-10-30: <a href="/person/1857/">N. Soave</a>.
<p>I will present some results regarding sharp a priori bounds, Liouville-type theorems, monotonicity and $1$-dimensional symmetry for solutions of the two systems
</p>
<p>
$
-\Delta u =u-u^3-\Lambda uv^2$ in $\mathbb{R}^N;$
$
-\Delta v =v-v^3-\Lambda u^2v$ in $\mathbb{R}^N$,
$u,v \ge 0$ in $\mathbb{R}^N$, with $\Lambda > 0$,
</p>
<p>and
</p>
<p>$
-\Delta u \, =-uv^2$ in $\mathbb{R}^N;$
$
-\Delta v \, =-u^2v$ in $\mathbb{R}^N$,
$u,v \ge 0$ in $\mathbb{R}^N$,
</p>
<p>
under suitable assumptions on the behavior of $u$ and $v$ when $x_N \to \pm \infty$. The talk is based on joint works with Alberto Farina, Berardino Sciunzi and Susanna Terracini.</p>
http://cvgmt.sns.it/seminar/598/