Groundwater flow in a fissurised porous stratum
Michiel Bertsch Carlo Nitsch
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.
keywords: Groundwater flow system of partial differential equations free boundary. fissurized porous medium
On a conjectured reverse Faber-Krahn inequality for a Steklov--type Laplacian eigenvalue
Vincenzo Ferone Carlo Nitsch Cristina Trombetti
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.
keywords: isoperimetric inequalities for eigenvalues weighted isoperimetric inequalities. Sharp trace embeddings
Shape optimization for Monge-Ampère equations via domain derivative
Barbara Brandolini Carlo Nitsch Cristina Trombetti
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.
keywords: Monge-Ampère equation domain derivative affine isoperimetric inequalities

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