Behaviour of $p$--Laplacian problems with Neumann boundary conditions when $p$ goes to 1
Anna Mercaldo Julio D. Rossi Sergio Segura de León Cristina Trombetti
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.
keywords: $1$--Laplacian. Neumann problem $p$--Laplacian
Hardy type inequalities and Gaussian measure
Barbara Brandolini Francesco Chiacchio Cristina Trombetti
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.
keywords: degenerate elliptic P.D.E.'s. Gaussian symmetrization Hardy type inequalities
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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