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CPAA

We consider the solution $u_p$ to the Neumann problem for the
$p$--Laplacian equation with the normal component of the flux
across the boundary given by $g\in L^\infty(\partial\Omega)$. We
study the behaviour of $u_p$ as $p$ goes to $1$ showing that they
converge to a measurable function $u$ and the gradients $|\nabla
u_p|^{p-2}\nabla u_p$ converge to a vector field $z$.
We prove that $z$ is bounded and that the properties of $u$ depend
on the size of $g$ measured in a suitable norm: if $g$ is small
enough, then $u$ is a function of bounded variation (it vanishes
on the whole domain, when $g$ is very small) while if $g$ is large
enough, then $u$ takes the value $\infty$ on a set of positive
measure. We also prove that in the first case, $u$ is a solution
to a limit problem that involves the $1-$Laplacian. Finally,
explicit examples are shown.

CPAA

In this paper we prove some improved Hardy type inequalities with
respect to the Gaussian measure. We show that they are strictly
related to the well-known Gross Logarithmic Sobolev inequality.
Some applications to elliptic P.D.E.'s are also given.

CPAA

For a given bounded Lipschitz set $\Omega$, we consider a Steklov--type eigenvalue problem for the Laplacian operator whose solutions provide extremal functions for the compact embedding $H^1(\Omega)\hookrightarrow L^2(\partial \Omega)$. We prove that a conjectured reverse Faber--Krahn inequality holds true at least in the class of Lipschitz sets which are ``close'' to a ball in a Hausdorff metric sense. The result implies that among sets of prescribed measure, balls are local minimizers of the embedding constant.

DCDS-S

In this note we prove that, if $\Omega$ is a smooth, strictly convex, open set in $R^n$ $(n \ge 2)$ with given measure, the $L^1$ norm of the convex solution to the Dirichlet problem $\det D^2 u=1$ in $\Omega$, $u=0$ on $\partial\Omega$, is minimum whenever $\Omega$ is an ellipsoid.

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