Radial Fast Diffusion on the Hyperbolic Space

We consider radial solutions to the fast diffusion equation $u_t=\Delta u^m$ on the hyperbolic space $\mathbb{H}^{N}$ for $N \ge 2$, $m\in(m_s,1)$, $m_s=\frac{N-2}{N+2}$. By radial we mean solutions depending only on the geodesic distance $r$ from a given point $o \in \mathbb{H}^N$. We investigate their fine asymptotics near the extinction time $T$ in terms of a separable solution of the form ${\mathcal V}(r,t)=(1-t/T)^{1/(1-m)}V^{1/m}(r)$, where $V$ is the unique positive energy solution, radial w.r.t. $o$, to $-\Delta V=c\,V^{1/m}$ for a suitable $c>0$, a semilinear elliptic problem thoroughly studied in \cite{MS08}, \cite{BGGV}. We show that $u$ converges to ${\mathcal V}$ in relative error, in the sense that $\|{u^m(\cdot,t)}/{{\mathcal V}^m(\cdot,t)}-1\|_\infty\to0$ as $t\to T^-$. In particular the solution is bounded above and below, near the extinction time $T$, by multiples of $(1-t/T)^{1/(1-m)}e^{-(N-1)r/m}$.Comment: To appear in Proc. London Math. So

Partial Differential Equations

The workshop dealt with partial differential equations in geometry and technical applications. The main topics were the combination of nonlinear partial differential equations and geometric problems, and fourth order equations in conformal geometry

Il metodo dei piani mobili: un approccio quantitativo

We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some recent results related to quantitative studies of the method of moving planes, where quantitative approximate symmetry results are obtained.Rivisitiamo risultati classici nei quali il metodo dei piani mobili è stato usato per dimostrare proprietà di simmetria per problemi sovradeterminati (come il Teorema di Serrin) e per problemi di rigidità in analisi geometrica (come il Teorema di Alexandrov). Inoltre forniamo un panoramica di recenti risultati legati a studi quantitativi del metodo dei piani mobili, nei quali vengono dimostrati risultati di simmetria approssimata

Boundary value problems with measures for elliptic equations with singular potentials

We study the boundary value problem with Radon measures for nonnegative solutions of $L_Vu:=-\Delta u+Vu=0$ in a bounded smooth domain \Gw, when $V$ is a locally bounded nonnegative function. Introducing some specific capacity, we give sufficient conditions on a Radon measure \gm on \prt\Gw so that the problem can be solved. We study the reduced measure associated to this equation as well as the boundary trace of positive solutions. In the appendix A. Ancona solves a question raised by M. Marcus and L. V\'eron concerning the vanishing set of the Poisson kernel of $L_V$ for an important class of potentials $V$.Comment: Contient un Appendice d'A. Ancona intitul\'e A necessary condition for the fine regularity of a boundary point with respect to a Schr\"odinger equatio

Layer solutions for the fractional Laplacian on hyperbolic space

The workshop dealt with partial differential equations in ge om- etry and technical applications. The main topics were the co mbination of nonlinear partial differential equations and geometric pro blems, and fourth order equations in conformal geometry.Postprint (published version