1. CMB Online first
 CruzUribe, David; Rodney, Scott; Rosta, Emily

PoincarÃ© Inequalities and Neumann Problems for the $p$Laplacian
We prove an equivalence between weighted PoincarÃ© inequalities
and
the existence of weak solutions to a Neumann problem related
to a
degenerate $p$Laplacian. The PoincarÃ© inequalities are
formulated in the context of degenerate Sobolev spaces defined
in
terms of a quadratic form, and the associated matrix is the
source of
the degeneracy in the $p$Laplacian.
Keywords:degenerate Sobolev space, $p$Laplacian, PoincarÃ© inequalities Categories:30C65, 35B65, 35J70, 42B35, 42B37, 46E35 

2. CMB 2007 (vol 50 pp. 35)
 Duyckaerts, Thomas

A Singular Critical Potential for the SchrÃ¶dinger Operator
Consider a real potential $V$ on
$\RR^d$, $d\geq 2$, and the Schr\"odinger equation:
\begin{equation}
\tag{LS} \label{LS1} i\partial_t u +\Delta u Vu=0,\quad
u_{\restriction t=0}=u_0\in L^2.
\end{equation}
In this paper, we investigate the minimal local regularity of $V$
needed to get local in time dispersive estimates (such as local in
time Strichartz estimates or local smoothing effect with gain of
$1/2$ derivative) on solutions of \eqref{LS1}. Prior works
show some dispersive properties when $V$ (small at infinity) is in
$L^{d/2}$ or in spaces just a little larger but with a smallness
condition on $V$ (or at least on its negative part).
In this work, we prove the critical character of these results by
constructing a positive potential $V$ which has compact support,
bounded outside $0$ and of the order $(\logx)^2/x^2$ near $0$.
The lack of dispersiveness comes from the existence of a sequence
of quasimodes for the operator $P:=\Delta+V$.
The elementary construction of $V$ consists in sticking together
concentrated, truncated potential wells near $0$. This yields a
potential oscillating with infinite speed and amplitude at $0$,
such that the operator $P$ admits a sequence of quasimodes of
polynomial order whose support concentrates on the pole.
Categories:35B65, 35L05, 35Q40, 35Q55 
