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Paolo Salani

Publications and source records attributed to Paolo Salani.

32 records · Page 2Linked to original sources
arXiv2019

Parabolic Minkowski convolutions of viscosity solutions to fully nonlinear equations

This paper is concerned with the Minkowski convolution of viscosity solutions of fully nonlinear parabolic equations. We adopt this convolution to compare viscosity solutions of initial-boundary value problems in different domains. As a consequence, we can for instance obtain parabolic power concavity of solutions to a general class of parabolic equations. Our results apply to the Pucci operator, the normalized $q$-Laplacians with $1<q\leq\infty$, the Finsler Laplacian and more general quasilinear operators.

math.AP↗
arXiv2019

To logconcavity and beyond

In 1976 Brascamp and Lieb proved that the heat flow preserves logconcavity. In this paper, introducing a variation of concavity, we show that it preserves in fact a stronger property than logconcavity and we identify the strongest concavity preserved by the heat flow.

math.AP↗
arXiv2018

Starshapedeness for fully-nonlinear equations in Carnot groups

In this paper we establish the starshapedness of the level sets of the capacitary potential of a large class of fully-nonlinear equations for condensers in Carnot groups, once a natural notion of starshapedness has been introduced. Our main result is Theorem 1.2 below.

math.AP↗
arXiv2018

Starshape of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings

In the present work we study solutions of the problem $-(-Δ)^{α/2}u = f(x,u)$ in $D_0\setminus \overline{D}_1$, with exterior conditions $u = 0$ in $R^N \setminus D_0$ and $u = 1$ in $\overline{D}_1$, where $D_1, D_0 \subset R^N$ are open sets such that $\overline{D}_1 \subset D_0$, $α\in (0,2)$, and $f$ is a nonlinearity. Under different assumptions on $f$ we prove that, if $D_0$ and $D_1$ are starshaped with respect to the same point $\bar{x} \in \overline{D}_1$, then the same occurs for every superlevel set of $u$.

math.AP↗
arXiv2018

Some overdetermined problems related to the anisotropic capacity

We characterize the Wulff shape of an anisotropic norm in terms of solutions to overdetermined problems for the Finsler $p$-capacity of a convex set $Ω\subset \mathbb{R}^N$, with $1<p<N$. In particular we show that if the Finsler $p$-capacitary potential $u$ associated to $Ω$ has two homothetic level sets then $Ω$ is Wulff shape. Moreover, we show that the concavity exponent of $u$ is $q=-(p-1)/(N-p)$ if and only if $Ω$ is Wulff shape.

math.AP↗
arXiv2017

Stability for Borell-Brascamp-Lieb inequalities

We study stability issues for the so-called Borell-Brascamp-Lieb inequalities, proving that when near equality is realized, the involved functions must be $L^1$-close to be $p$-concave and to coincide up to homotheties of their graphs.

math.FA↗
arXiv2016

An overdetermined problem for the anisotropic capacity

We consider an overdetermined problem for the Finsler Laplacian in the exterior of a convex domain in $\mathbb{R}^N$, establishing a symmetry result for the anisotropic capacitary potential. Our result extends the one of W. Reichel [Arch. Rational Mech. Anal. 137 (1997)], where the usual Newtonian capacity is considered, giving rise to an overdetermined problem for the standard Laplace equation. Here, we replace the usual Euclidean norm of the gradient with an arbitrary norm $H$. The resulting symmetry of the solution is that of the so-called Wulff shape (a ball in the dual norm $H_0$).

math.AP↗
arXiv2013

Parabolic power concavity and parabolic boundary value problems

This paper is concerned with power concavity properties of the solution to the parabolic boundary value problem \begin{equation} \tag{$P$} \left\{\begin{array}{ll} \partial_t u=Δu +f(x,t,u,\nabla u) & \mbox{in}\quadΩ\times(0,\infty),\vspace{3pt}\\ u(x,t)=0 & \mbox{on}\quad\partial Ω\times(0,\infty),\vspace{3pt}\\ u(x,0)=0 & \mbox{in}\quadΩ, \end{array} \right. \end{equation} where $Ω$ is a bounded convex domain in ${\bf R}^n$ and $f$ is a nonnegative continuous function in $Ω\times(0,\infty)\times{\bf R}\times{\bf R}^n$. We give a sufficient condition for the solution of $(P)$ to be parabolically power concave in $\barΩ\times[0,\infty)$.

math.AP↗
arXiv2013

An overdetermined problem with non constant boundary condition

We investigate an overdetermined Torsion problem, with a non-constant positively homogeneous boundary constraint on the gradient. We interpret this problem as the Euler equation of a shape optimization problems, we prove existence and regularity of a solution. Moreover several geometric properties of the solution are shown.

math.AP↗
arXiv2010

The location of the hot spot in a grounded convex conductor

We investigate the location of the (unique) hot spot in a convex heat conductor with unitary initial temperature and with boundary grounded at zero temperature. We present two methods to locate the hot spot: the former is based on ideas related to the Alexandrov-Bakelmann-Pucci maximum principle and Monge-Ampère equations; the latter relies on Alexandrov's reflection principle. We then show how such a problem can be simplified in case the conductor is a polyhedron. Finally, we present some numerical computations.

math.AP↗