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
Communications on Pure & Applied Analysis 2013, 12(1): 253-267 doi: 10.3934/cpaa.2013.12.253
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
Shape optimization for Monge-Ampère equations via domain derivative
Barbara Brandolini Carlo Nitsch Cristina Trombetti
Discrete & Continuous Dynamical Systems - S 2011, 4(4): 825-831 doi: 10.3934/dcdss.2011.4.825
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
Hardy type inequalities and Gaussian measure
Barbara Brandolini Francesco Chiacchio Cristina Trombetti
Communications on Pure & Applied Analysis 2007, 6(2): 411-428 doi: 10.3934/cpaa.2007.6.411
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
Communications on Pure & Applied Analysis 2015, 14(1): 63-82 doi: 10.3934/cpaa.2015.14.63
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

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