19 citations · 69 across the 31 of their papers we have counts for
35 papers
A Brunn--Minkowski inequality and Convexity for the 2-Hessian eigenvalue in convex domains
Jiahuan Li, Xi-Nan Ma, Guohuan Qiu +1
We prove the strict log-concavity of the positive first eigenfunction \(-u\) of the \(2\)-Hessian equation and the strict -convexity of the solution for the corresponding tors…
A Brunn--Minkowski inequality for the Hessian eigenvalue in convex domain
Jiahuan Li, Xi-Nan Ma, Paolo Salani
We use the deformation methods to obtain the strictly log concavity of solution of a class Hessian equation in bounded convex domain in , as an application we get t…
Preservation of F-convexity under the heat flow
Kazuhiro Ishige, Troy Petitt, Paolo Salani
We introduce the notion of F-convexity as a general extension of power convexity. We characterize the F-convexities preserved under the heat flow in the n-dimensional Euclidean spa…
Log-concavity of solutions of parabolic equations related to the Ornstein-Uhlenbeck operator and applications
Andrea Colesanti, Lei Qin, Paolo Salani
In this paper, we investigate the log-concavity of the kernel for the parabolic Ornstein-Uhlenbeck operator in a bounded, convex domain. Consequently, we get the preservation of th…
A constant rank theorem for linear elliptic equations on the sphere with applications to the mixed Christoffel problem
A. Colesanti, M. Focardi, P. Guan +1
We study the mixed Christoffel problem for convex bodies providing sufficient conditions for its solution. Key to our approach is a constant rank theorem, following the a…
Geometric properties of solutions to elliptic PDE's in Gauss space and related Brunn-Minkowski type inequalities
Andrea Colesanti, Lei Qin, Paolo Salani
We prove a Brunn-Minkowski type inequality for the first (nontrivial) Dirichlet eigenvalue of the weighted -operator \[ -Δ_{p,γ}u=-\text{div}(|\nabla u|^{p-2} \nabla u)+(x,\nabl…
Qualitative properties of free boundaries for the exterior Bernoulli problem for the half Laplacian
Sven Jarohs, Tadeusz Kulczycki, Paolo Salani
In this work, we study the asymptotic behavior of the free boundary of the solution to the exterior Bernoulli problem for the half Laplacian when the Bernoulli's gradient parameter…
Riemannian starshape and capacitary problems
Kazuhiro Ishige, Paolo Salani, Asuka Takatsu
We prove the Riemannian version of a classical Euclidean result: every level set of the capacitary potential of a starshaped ring is starshaped. In the Riemannian setting, we restr…
The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction
Andrea Colesanti, Elisa Francini, Galyna Livshyts +1
We prove that the first (nontrivial) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator as a function of the domain, is convex with r…
A parabolic PDE-based approach to Borell--Brascamp--Lieb inequality
Kazuhiro Ishige, Qing Liu, Paolo Salani
In this paper, we provide a new PDE proof for the celebrated Borell--Brascamp--Lieb inequality. Our approach reveals a deep connection between the Borell--Brascamp--Lieb inequality…
Positive solutions of a critical equation in sub-Finsler geometry
Nicola Garofalo, Paolo Salani
We compute a two-parameter family of explicit positive solutions of a critical Yamabe type equation for a nonlinear operator that sits at the intersection of Finsler and sub-Rieman…
A fundamental solution for a subelliptic operator in Finsler geometry
Federica Dragoni, Nicola Garofalo, Gianmarco Giovannardi +1
We introduce a class of nonlinear partial differential equations in a product space which are at the interface of Finsler and sub-Riemannian geometry. To such equations we associat…
Wulff shape symmetry of solutions to overdetermined problems for Finsler Monge-Ampère equations
Andrea Cianchi, Paolo Salani
We deal with Monge-Ampère type equations modeled upon general anisotropic norms in . An overdetermined problem for convex solutions to these equations is analyzed.…
Characterization of -concavity preserved by the Dirichlet heat flow
Kazuhiro Ishige, Paolo Salani, Asuka Takatsu
-concavity is a generalization of power concavity and, actually, the largest available generalization of the notion of concavity. We characterize the -concavities preserved b…
On the Bernoulli free boundary problems for the half Laplacian and for the spectral half Laplacian
Sven Jarohs, Tadeusz Kulczycki, Paolo Salani
We study the exterior and interior Bernoulli problems for the half Laplacian and the interior Bernoulli problem for the spectral half Laplacian. We concentrate on the existence and…
Power concavity and Dirichlet heat flow
Kazuhiro Ishige, Paolo Salani, Asuka Takatsu
We show that log-concavity is the weakest power concavity preserved by the Dirichlet heat flow in -dimensional convex domains, where (indeed, we prove that starting wit…
New characterizations of log-concavity
Kazuhiro Ishige, Paolo Salani, Asuka Takatsu
We introduce a notion of -concavity which largely generalizes the usual concavity. By the use of the notions of closedness under positive scalar multiplication and closedness un…
Power concavity for elliptic and parabolic boundary value problems on rotationally symmetric domains
Kazuhiro Ishige, Paolo Salani, Asuka Takatsu
We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applica…
Parabolic Minkowski convolutions of viscosity solutions to fully nonlinear equations
Kazuhiro Ishige, Qing Liu, Paolo Salani
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 in…
To logconcavity and beyond
Kazuhiro Ishige, Paolo Salani, Asuka Takatsu
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 proper…
Starshapedeness for fully-nonlinear equations in Carnot groups
Federica Dragoni, Nicola Garofalo, Paolo Salani
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 na…
Some overdetermined problems related to the anisotropic capacity
Chiara Bianchini, Giulio Ciraolo, Paolo Salani
We characterize the Wulff shape of an anisotropic norm in terms of solutions to overdetermined problems for the Finsler -capacity of a convex set , with $…
Optimal concavity of the torsion function
A. Henrot, C. Nitsch, P. Salani +1
In this short note we consider an unconventional overdetermined problem for the torsion function: let and be a bounded open set in whose torsion functi…
Starshape of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings
Sven Jarohs, Tadeusz Kulczycki, Paolo Salani
In the present work we study solutions of the problem in , with exterior conditions in and …