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### Open Access Journals

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.

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.

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.

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