arXiv subjects

P. Salani

Publications and source records attributed to P. Salani.

3 recordsLinked to original sources

arXiv2025

A constant rank theorem for linear elliptic equations on the sphere with applications to the mixed Christoffel problem

We study the mixed Christoffel problem for $C^{2,+}$ convex bodies providing sufficient conditions for its solution. Key to our approach is a constant rank theorem, following the approach developed in \cite{Guan-Ma-2003} to address the Christoffel problem, in order to ensure that the solution to a related second order linear PDE on the sphere is indeed geometric, that is, it is the support functions of a $C^{2,+}$ convex body.

math.AP↗

arXiv2017

Optimal concavity of the torsion function

In this short note we consider an unconventional overdetermined problem for the torsion function: let $n\geq 2$ and $Ω$ be a bounded open set in $\mathbb{R}^n$ whose torsion function $u$ (i.e. the solution to $Δu=-1$ in $Ω$, vanishing on $\partialΩ$) satisfies the following property: $\sqrt{M-u(x)}$ is convex, where $M=\max\{u(x)\,:\,x\in\overlineΩ\}$. Then $Ω$ is an ellipsoid.

math.AP↗

arXiv2010

Concavity properties for free boundary elliptic problems

We prove some concavity properties connected to nonlinear Bernoulli type free boundary problems. In particular, we prove a Brunn-Minkowski inequality and an Urysohn's type inequality for the Bernoulli Constant and we study the behaviour of the free boundary with respect to the given boundary data. Moreover we prove a uniqueness result regarding the interior non-linear Bernoulli problem.

math.AP↗