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NHM

In [2] Barenblatt e.a. introduced a fluid model
for groundwater flow in fissurised porous media.
The system consists of two diffusion equations for the groundwater
levels in, respectively, the porous bulk and the system of cracks.
The equations are coupled by a fluid exchange term.
Numerical evidence in [2, 8] suggests that the penetration depth of the fluid
increases dramatically due to the presence of cracks and that the smallness of
certain parameter values play a key role in this phenomenon.
In the present paper we give precise estimates for the
penetration depth in terms of the smallness of some of the parameters.

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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