Paolo Salani

We investigate the convexity of level sets of solutions to general elliptic equations in a convex ring Ω. In particular, if u is a classical solution which has constant (distinct) … We investigate the convexity of level sets of solutions to general elliptic equations in a convex ring Ω. In particular, if u is a classical solution which has constant (distinct) values on the two connected components of ∂Ω, we consider its quasi-concave envelope u* (i.e., the function whose superlevel sets are the convex envelopes of those of u) and we find suitable assumptions which force u * to be a subsolution of the equation. If a comparison principle holds, this yields u = u* and then u is quasi-concave.

Quasi–concave envelope of a function and convexity of level sets of solutions to elliptic equations

2003-08-14

Andrea Colesanti , Paolo Salani

Abstract Given a C 2 function u , we consider its quasi–convex envelope u * and we investigate the relationship between D 2 u and D 2 u * (the … Abstract Given a C 2 function u , we consider its quasi–convex envelope u * and we investigate the relationship between D 2 u and D 2 u * (the latter intended in viscosity sense); we obtain two inequalities between the tangential Laplacian of u and u * and the normal second derivative of u and u * (the words tangential and normal are referred to a level set of the involved functions). Then we apply the result to prove convexity of level sets of solutions of elliptic equations in convex rings. Our results can be applied to a class of elliptic operator which can be naturally decomposed in a tangential and a normal part, such as Laplacian, p –Laplacian or the Mean Curvature operator. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An overdetermined problem for the anisotropic capacity

2016-06-25

Chiara Bianchini , Giulio Ciraolo , Paolo Salani

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On the Hessian matrix and Minkowski addition of quasiconvex functions

2007-07-05

Marco Longinetti , Paolo Salani

The location of the hot spot in a grounded convex conductor

2011-01-01

Lorenzo Brasco , Rolando Magnanini , Paolo Salani

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 … 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.

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Convexity and asymptotic estimates for large solutions of Hessian equations

2000-01-01

Andrea Colesanti , Elisa Francini , Paolo Salani

We consider the smallest viscosity solution of the Hessian equation $S_k(D^2u)=f(u)$ in $\omega$, that becomes infinite at the boundary of $\omega\subset\mathbb R^n$; here $S_k(D^2u)$ denotes the $k$-th elementary symmetric function … We consider the smallest viscosity solution of the Hessian equation $S_k(D^2u)=f(u)$ in $\omega$, that becomes infinite at the boundary of $\omega\subset\mathbb R^n$; here $S_k(D^2u)$ denotes the $k$-th elementary symmetric function of the eigenvalues of $D^2u$, for $k\in\{1,\dots, n\}$. We prove that if $\omega$ is strictly convex and $f$ satisfies suitable assumptions, then the smallest solution is convex. We also establish asymptotic estimates for the behaviour of such a solution near the boundary of $\omega$.

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Is quasi-concavity preserved by heat flow?

2008-04-16

Kazuhiro Ishige , Paolo Salani

Parabolic quasi‐concavity for solutions to parabolic problems in convex rings

2010-09-22

Kazuhiro Ishige , Paolo Salani

Abstract We investigate some geometric properties of level sets of the solutions of parabolic problems in convex rings. We introduce the notion of parabolic quasi‐concavity , which involves time and … Abstract We investigate some geometric properties of level sets of the solutions of parabolic problems in convex rings. We introduce the notion of parabolic quasi‐concavity , which involves time and space jointly and is a stronger property than the spatial quasi‐concavity, and study the convexity of superlevel sets of the solutions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Convexity of level sets for solutions to nonlinear elliptic problems in convex rings

2006-10-01

Paola Cuoghi , Paolo Salani

We find suitable assumptions for the quasi-concave envelope u of a solution (or a subsolution) u of an elliptic equation F(x,u,ru,D 2 u) = 0 (possibly fully nonlinear) to be … We find suitable assumptions for the quasi-concave envelope u of a solution (or a subsolution) u of an elliptic equation F(x,u,ru,D 2 u) = 0 (possibly fully nonlinear) to be a viscosity subsolution of the same equation. We apply this result to study the convexity of level sets of solutions to elliptic Dirichlet problems in a convex ring = 0 1.

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A Brunn–Minkowski inequality for the Monge–Ampère eigenvalue

2004-07-21

Paolo Salani

Brunn-Minkowski inequalities for two functionals involving the<b><i>p</i></b>-Laplace operator

2006-01-01

Andrea Colesanti , Paola Cuoghi , Paolo Salani

In the family of n-dimensional convex bodies, we prove a Brunn-Minkowski type inequality for the first eigenvalue of the p-Laplace operator, or Poincaré constant, and for a functional extending the … In the family of n-dimensional convex bodies, we prove a Brunn-Minkowski type inequality for the first eigenvalue of the p-Laplace operator, or Poincaré constant, and for a functional extending the notion of torsional rigidity. In the latter case we also characterize equality conditions.

Convexity of solutions and Brunn–Minkowski inequalities for Hessian equations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:math>

2011-12-30

Paolo Salani

On a new kind of convexity for solutions of parabolic problems

2010-12-08

Kazuhiro Ishige , Paolo Salani

We introduce the notion of $\alpha$-parabolic quasi-concavity for functions of space and time, which extends the usual notion of quasi-concavity and the notion of parabolic quasi-cocavity introduced in [18]. Then … We introduce the notion of $\alpha$-parabolic quasi-concavity for functions of space and time, which extends the usual notion of quasi-concavity and the notion of parabolic quasi-cocavity introduced in [18]. Then we investigate the $\alpha$-parabolic quasi-concavity of solutions to parabolic problems with vanishing initial datum. The results here obtained are generalizations of some of the results of [18].

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Generalised solutions of Hessian equations

1997-12-01

Andrea Colesanti , Paolo Salani

We introduce a definition of generalised solutions of the Hessian equation S m ( D 2 u ) = f in a convex set ω ⊂ ℝ n , where … We introduce a definition of generalised solutions of the Hessian equation S m ( D 2 u ) = f in a convex set ω ⊂ ℝ n , where S m ( D 2 u ) denotes the m -th symmetric function of the eigenvalues of D 2 u , f ∈ L p (ω), p ≥ 1, and m ∈ {1, …, n }. Such a definition is given in the class of semi-convex functions, and it extends the definition of convex generalised solutions for the Monge-Ampère equation. We prove that semiconvex weak solutions are solutions in the sense of the present paper.

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Starshapedness of level sets of solutions to elliptic PDEs

2005-10-06

Paolo Salani

We investigate how some geometric properties of the domain are inherited by level sets of solutions of elliptic equations. In particular we prove that, under suitable assumptions, solutions of elliptic … We investigate how some geometric properties of the domain are inherited by level sets of solutions of elliptic equations. In particular we prove that, under suitable assumptions, solutions of elliptic Dirichlet problems in starshaped rings have starshaped level sets. Our results are applicable to a large class of operators, including fully-nonlinear ones.

An overdetermined problem with non-constant boundary condition

2014-07-09

Chiara Bianchini , Antoine Henrot , Paolo Salani

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 problem, we … 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 problem, we prove existence and regularity of a solution. Moreover several geometric properties of the solution are shown.

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Parabolic power concavity and parabolic boundary value problems

2013-12-04

Kazuhiro Ishige , Paolo Salani

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Concavity properties for elliptic free boundary problems

2009-03-10

Chiara Bianchini , Paolo Salani

Hessian equations in non-smooth domains

1999-12-01

Andrea Colesanti , Paolo Salani

Convexity breaking of the free boundary for porous medium equations

2010-02-25

Kazuhiro Ishige , Paolo Salani

We investigate the preservation of convexity of the free boundary by the solutions of the porous medium equation. We prove that starting with an initial datum with some kind of … We investigate the preservation of convexity of the free boundary by the solutions of the porous medium equation. We prove that starting with an initial datum with some kind of suboptimal αconcavity property, the convexity of the positivity set can be lost in a short time.

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Stability of radial symmetry for a Monge-Ampère overdetermined problem

2008-09-16

Barbara Brandolini , Carlo Nitsch , Paolo Salani +1 more

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Geometric Properties for Parabolic and Elliptic PDE's

2016-01-01

Filippo Gazzola , Kazuhiro Ishige , Carlo Nitsch +1 more

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Stability for a strengthened Borell–Brascamp–Lieb inequality

2018-03-21

A.P. Rossi , Paolo Salani

We study stability issues for some integral inequalities, in particular for the extremal case of the so-called Borell–Brascamp–Lieb inequalities. In this case, when near equality is realized in dimension one, … We study stability issues for some integral inequalities, in particular for the extremal case of the so-called Borell–Brascamp–Lieb inequalities. In this case, when near equality is realized in dimension one, we prove that the involved functions must be L1-close to be quasiconcave. Further results about equality conditions in some other cases are provided.

Power Concavity for Solutions of Nonlinear Elliptic Problems in Convex Domains

2012-11-26

Massimiliano Bianchini , Paolo Salani

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A characterization of balls through optimal concavity for potential functions

2014-08-28

Paolo Salani

In this short note two unconventional overdetermined problems are considered. Let $p\in (1,n)$; first, the following is proved: if $\Omega$ is a bounded domain in $\mathbb {R}^n$ whose $p$-capacitary potential … In this short note two unconventional overdetermined problems are considered. Let $p\in (1,n)$; first, the following is proved: if $\Omega$ is a bounded domain in $\mathbb {R}^n$ whose $p$-capacitary potential function $u$ has two homotetic convex level sets, then $\Omega$ is a ball. Then, as an application, we obtain the following: if $\Omega$ is a convex domain in $\mathbb {R}^n$ whose $p$-capacitary potential function $u$ is $(1-p)/(n-p)$-concave (i.e. $u^{(1-p)/(n-p)}$ is convex), then $\Omega$ is a ball.

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Optimal Concavity of the Torsion Function

2018-05-16

Antoine Henrot , Carlo Nitsch , Paolo Salani +1 more

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A note on parabolic power concavity

2014-10-01

Kazuhiro Ishige , Paolo Salani

We investigate parabolic power concavity properties of the solutions of the heat equation in W Â ½0; TÞ, where W ¼ R n or W is a bounded convex domain … We investigate parabolic power concavity properties of the solutions of the heat equation in W Â ½0; TÞ, where W ¼ R n or W is a bounded convex domain in R n .

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Combination and mean width rearrangements of solutions to elliptic equations in convex sets

2014-04-16

Paolo Salani

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Power concavity in weakly coupled elliptic and parabolic systems

2015-09-12

Kazuhiro Ishige , Kazushige Nakagawa , Paolo Salani

To logconcavity and beyond

2019-01-22

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 property … 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.

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A proof of a Loomis–Whitney type inequality via optimal transport

2018-10-29

Stefano Campi , Paolo Gronchi , Paolo Salani

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Starshapedeness for fully non‐linear equations in Carnot groups

2018-12-04

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. 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.

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Stability for Borell-Brascamp-Lieb Inequalities

2017-01-01

A.P. Rossi , Paolo Salani

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On Space-Time Quasiconcave Solutions of the Heat Equation

2019-04-12

Chuanqiang Chen , Xi‐Nan Ma , Paolo Salani

In this paper we first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing … In this paper we first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing this constant rank theorem, we can obtain some strictly convexity results of the spatial and space-time level sets of the space-time quasiconcave solution of the heat equation in a convex ring. To explain our ideas and for completeness, we also review the constant rank theorem technique for the space-time Hessian of space-time convex solution of heat equation and for the second fundamental form of the convex level sets for harmonic function.

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Convexity of solutions and Brunn-Minkowski inequalities for Hessian equations in $\R^3$

2011-01-01

Paolo Salani

By using Minkowski addition of convex functions, we prove convexity and rearrangement properties of solutions to some Hessian equations in $\R^3$ and Brunn-Minkowski and isoperimetric inequalities for related functionals. By using Minkowski addition of convex functions, we prove convexity and rearrangement properties of solutions to some Hessian equations in $\R^3$ and Brunn-Minkowski and isoperimetric inequalities for related functionals.

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Parabolic Minkowski convolutions of solutions to parabolic boundary value problems

2015-11-20

Kazuhiro Ishige , Paolo Salani

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Parabolic Minkowski convolutions and concavity properties of viscosity solutions to fully nonlinear equations

2019-12-09

Kazuhiro Ishige , Qing Liu , Paolo Salani

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Spatial concavity of solutions to parabolic systems

2018-04-09

Kazuhiro Ishige , Kazushige Nakagawa , Paolo Salani

We investigate spatial log-concavity and spatial power concavity of solutions to parabolic systems with log-concave or power concave initial data in convex domains. We investigate spatial log-concavity and spatial power concavity of solutions to parabolic systems with log-concave or power concave initial data in convex domains.

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Geometric Properties for Parabolic and Elliptic PDE's

2021-01-01

Vincenzo Ferone , Tatsuki Kawakami , Paolo Salani +1 more

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On space-time quasiconcave solutions of the heat equation

2014-01-01

Chuanqiang Chen , Xi‐Nan Ma , Paolo Salani

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Stability of isoperimetric type inequalities for some Monge–Ampère functionals

2012-10-11

Daria Ghilli , Paolo Salani

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Parabolic Minkowski convolutions of viscosity solutions to fully nonlinear equations

2019-01-01

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 initial-boundary value problems in different … 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.

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Some overdetermined problems related to the anisotropic capacity

2018-05-04

Chiara Bianchini , Giulio Ciraolo , Paolo Salani

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Power concavity for elliptic and parabolic boundary value problems on rotationally symmetric domains

2021-10-25

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 applications of our results to the hyperbolic … We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applications of our results to the hyperbolic space [Formula: see text], we have The first (positive) Dirichlet eigenfunction of the Laplacian on a ball in [Formula: see text] raised to some power [Formula: see text] is strictly concave; Let [Formula: see text] be the heat kernel on [Formula: see text]. Then [Formula: see text] is strictly log-concave in [Formula: see text] for [Formula: see text] and [Formula: see text].

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Riemannian starshape and capacitary problems

2025-03-23

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

Geometric properties of solutions to elliptic PDE's in Gauss space and related Brunn-Minkowski type inequalities

2025-01-31

Andrea Colesanti , Lei Qin , Paolo Salani

We prove a Brunn-Minkowski type inequality for the first (nontrivial) Dirichlet eigenvalue of the weighted $p$-operator \[ -\Delta_{p,\gamma}u=-\text{div}(|\nabla u|^{p-2} \nabla u)+(x,\nabla u)|\nabla u|^{p-2}, \] where $p>1$, in the class of … We prove a Brunn-Minkowski type inequality for the first (nontrivial) Dirichlet eigenvalue of the weighted $p$-operator \[ -\Delta_{p,\gamma}u=-\text{div}(|\nabla u|^{p-2} \nabla u)+(x,\nabla u)|\nabla u|^{p-2}, \] where $p>1$, in the class of bounded Lipschitz domains in $\mathbb{R}^n$. We also prove that any corresponding positive eigenfunction is log-concave if the domain is convex.

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Qualitative properties of free boundaries for the exterior Bernoulli problem for the half Laplacian.

2025-01-01

Sven Jarohs , Tadeusz Kulczycki , Paolo Salani

Positive solutions of a critical equation in sub-Finsler geometry

2024-10-04

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-Riemannian geometry. 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-Riemannian geometry.

Qualitative properties of free boundaries for the exterior Bernoulli problem for the half Laplacian

2024-09-26

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 tends … 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 tends to $0^+$ and to $+\infty$. Moreover, we show that, under suitable conditions, the perpendicular rays of the free boundary always meets the convex envelope of the fixed boundary.

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The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction

2024-07-31

Andrea Colesanti , Elisa Francini , Galyna V. Livshyts +1 more

We prove that the first (nontrivial) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator $$ L(u)=\Delta u-\langle\nabla u,x\rangle\,, $$ as a function of the domain, is convex with respect to the Minkowski … We prove that the first (nontrivial) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator $$ L(u)=\Delta u-\langle\nabla u,x\rangle\,, $$ as a function of the domain, is convex with respect to the Minkowski addition, and we characterize the equality cases in some classes of convex sets. We also prove that the corresponding (positive) eigenfunction is log-concave if the domain is convex.

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A parabolic PDE-based approach to Borell--Brascamp--Lieb inequality

2024-05-26

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 and properties of diffusion equations of … 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 and properties of diffusion equations of porous medium type pertaining to the large time asymptotics and preservation of a generalized concavity of the solutions. We also recover the equality condition in the special case of the Pr\'ekopa--Leindler inequality by further exploiting known properties of the heat equation including the eventual log-concavity and backward uniqueness of solutions.

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A fundamental solution for a subelliptic operator in Finsler geometry

2024-01-01

Federica Dragoni , Nicola Garofalo , Gianmarco Giovannardi +1 more

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 associate a non-isotropic … 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 associate a non-isotropic Minkowski gauge $\Theta$ for which we introduce a suitable notion of Legendre transform $\Theta^0$. We compute the action of the relevant nonlinear PDEs on ``radial" functions, i.e., functions of $\Theta^0$, and by exploiting it we are able to compute explicit fundamental solutions of such PDEs.

Positive solutions of a critical equation in sub-Finsler geometry

2024-01-01

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-Riemannian geometry 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-Riemannian geometry

Wulff shape symmetry of solutions to overdetermined problems for Finsler Monge-Ampère equations

2023-07-13

Andrea Cianchi , Paolo Salani

We deal with Monge-Ampère type equations modeled upon general Finsler norms H in Rn. An overdetermined problem for convex solutions to these equations is analyzed. The relevant solutions are subject … We deal with Monge-Ampère type equations modeled upon general Finsler norms H in Rn. An overdetermined problem for convex solutions to these equations is analyzed. The relevant solutions are subject to both a homogeneous Dirichlet condition and a second boundary condition, designed on H, on the gradient image of the domain. The Wulff shape symmetry associated with H of the solutions is established.

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Hierarchy of deformations in concavity

2022-12-12

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

On the Bernoulli free boundary problems for the half Laplacian and for the spectral half Laplacian

2022-05-10

Sven Jarohs , Tadeusz Kulczycki , Paolo Salani

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Characterization of $F$-concavity preserved by the Dirichlet heat flow

2022-01-01

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

$F$-concavity is a generalization of power concavity and, actually, the largest available generalization of the notion of concavity. We characterize the $F$-concavities preserved by the Dirichlet heat flow in convex … $F$-concavity is a generalization of power concavity and, actually, the largest available generalization of the notion of concavity. We characterize the $F$-concavities preserved by the Dirichlet heat flow in convex domains on ${\mathbb R}^n$, and complete the study of preservation of concavity properties by the Dirichlet heat flow, started by Brascamp and Lieb in 1976 and developed in some recent papers. More precisely: (1) we discover hot-concavity, which is the strongest $F$-concavity preserved by the Dirichlet heat flow; (2) we show that log-concavity is the weakest $F$-concavity preserved by the Dirichlet heat flow; quasi-concavity is also preserved only for $n=1$; (3) we prove that if $F$-concavity does not coincide with log-concavity and it is not stronger than log-concavity and $n\ge 2$, then there exists an $F$-concave initial datum such that the corresponding solution to the Dirichlet heat flow is not even quasi-concave, hence losing any reminiscence of concavity. Furthermore, we find a sufficient and necessary condition for $F$-concavity to be preserved by the Dirichlet heat flow. We also study the preservation of concavity properties by solutions of the Cauchy--Dirichlet problem for linear parabolic equations with variable coefficients and for nonlinear parabolic equations such as semilinear heat equations, the porous medium equation, and the parabolic $p$-Laplace equation.

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Wulff shape symmetry of solutions to overdetermined problems for Finsler Monge-Ampère equations

2022-01-01

Andrea Cianchi , Paolo Salani

We deal with Monge-Ampère type equations modeled upon general anisotropic norms $H$ in $\mathbb R^n$. An overdetermined problem for convex solutions to these equations is analyzed. The relevant solutions are … We deal with Monge-Ampère type equations modeled upon general anisotropic norms $H$ in $\mathbb R^n$. An overdetermined problem for convex solutions to these equations is analyzed. The relevant solutions are subject to both a homogeneous Dirichlet condition and a second boundary condition, designed on $H$, on the gradient image of the domain. The Wulff shape symmetry associated with $H$ of the solutions is established.

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On the Bernoulli free boundary problems for the half Laplacian and for the spectral half Laplacian

2021-12-10

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 geometric properties … 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 geometric properties of solutions. Our main results are the following. For the exterior Bernoulli problem for the half Laplacian, we show that under starshapedness assumptions on the data the free domain is starshaped. For the interior Bernoulli problem for the spectral half Laplacian, we show that under convexity assumptions on the data the free domain is convex and we prove a Brunn-Minkowski inequality for the Bernoulli constant. For Bernoulli problems for the half Laplacian we use a variational approach, whereas for Bernoulli problem for the spectral half Laplacian we use the Beurling method based on subsolutions.

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New characterizations of log-concavity via Dirichlet heat flow

2021-11-22

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

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Power concavity for elliptic and parabolic boundary value problems on rotationally symmetric domains

2021-10-25

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

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Power concavity and Dirichlet heat flow

2021-01-01

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

We show that log-concavity is the weakest power concavity preserved by the Dirichlet heat flow in $N$-dimensional convex domains, where $N\ge 2$ (indeed, we prove that starting with a negative … We show that log-concavity is the weakest power concavity preserved by the Dirichlet heat flow in $N$-dimensional convex domains, where $N\ge 2$ (indeed, we prove that starting with a negative power concave initial datum may result in losing immediately any reminiscence of concavity). Jointly with what we already know, i.e. that log-concavity is the strongest power concavity preserved by the Dirichlet heat flow, we see that log-concavity is indeed the only power concavity preserved by the Dirichlet heat flow.

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On the Bernoulli free boundary problems for the half Laplacian and for the spectral half Laplacian

2021-01-01

Sven Jarohs , Tadeusz Kulczycki , Paolo Salani

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Geometric Properties for Parabolic and Elliptic PDE's

2021-01-01

Vincenzo Ferone , Tatsuki Kawakami , Paolo Salani +1 more

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Power concavity for elliptic and parabolic boundary value problems on rotationally symmetric domains

2020-02-24

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 applications of our results to the hyperbolic … We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applications of our results to the hyperbolic space ${\bf H}^N$ we have: $\bullet$ The first Dirichlet eigenfunction on a ball in ${\bf H}^N$ is strictly positive power concave; $\bullet$ Let $\Gamma$ be the heat kernel on ${\bf H}^N$. Then $\Gamma(\cdot,y,t)$ is strictly log-concave on ${\bf H}^N$ for $y\in {\bf H}^N$ and $t>0$.

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Power concavity for elliptic and parabolic boundary value problems on rotationally symmetric domains

2020-01-01

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 applications of our results to the hyperbolic … We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applications of our results to the hyperbolic space ${\bf H}^N$ we have: $\bullet$ The first Dirichlet eigenfunction on a ball in ${\bf H}^N$ is strictly positive power concave; $\bullet$ Let $\Gamma$ be the heat kernel on ${\bf H}^N$. Then $\Gamma(\cdot,y,t)$ is strictly log-concave on ${\bf H}^N$ for $y\in {\bf H}^N$ and $t>0$.

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New characterizations of log-concavity

2020-01-01

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

We introduce a notion of $F$-concavity which largely generalizes the usual concavity. By the use of the notions of closedness under positive scalar multiplication and closedness under positive exponentiation we … We introduce a notion of $F$-concavity which largely generalizes the usual concavity. By the use of the notions of closedness under positive scalar multiplication and closedness under positive exponentiation we characterize power concavity and power log-concavity among nontrivial $F$-concavities, respectively. In particular, we have a characterization of log-concavity as the only $F$-concavity which is closed both under positive scalar multiplication and positive exponentiation. Furthermore, we discuss the strongest $F$-concavity preserved by the Dirichlet heat flow, characterizing log-concavity also in this connection.

Parabolic Minkowski convolutions and concavity properties of viscosity solutions to fully nonlinear equations

2019-12-09

Kazuhiro Ishige , Qing Liu , Paolo Salani

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On Space-Time Quasiconcave Solutions of the Heat Equation

2019-04-12

Chuanqiang Chen , Xi‐Nan Ma , Paolo Salani

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To logconcavity and beyond

2019-01-22

Kazuhiro Ishige , Paolo Salani , Asuka Takatsu

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

2019-01-17

Sven Jarohs , Tadeusz Kulczycki , Paolo Salani

Abstract We study solutions of the problem urn:x-wiley:0025584X:media:mana201700226:mana201700226-math-0001 where are open sets such that , , and f is a nonlinearity. Under different assumptions on f we prove that, if … Abstract We study solutions of the problem urn:x-wiley:0025584X:media:mana201700226:mana201700226-math-0001 where are open sets such that , , 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 , then the same occurs for every superlevel set of u .

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Parabolic Minkowski convolutions of viscosity solutions to fully nonlinear equations

2019-01-01

Kazuhiro Ishige , Qing Liu , Paolo Salani

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Starshapedeness for fully non‐linear equations in Carnot groups

2018-12-04

Federica Dragoni , Nicola Garofalo , Paolo Salani

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A proof of a Loomis–Whitney type inequality via optimal transport

2018-10-29

Stefano Campi , Paolo Gronchi , Paolo Salani

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Optimal Concavity of the Torsion Function

2018-05-16

Antoine Henrot , Carlo Nitsch , Paolo Salani +1 more

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Some overdetermined problems related to the anisotropic capacity

2018-05-04

Chiara Bianchini , Giulio Ciraolo , Paolo Salani

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Spatial concavity of solutions to parabolic systems

2018-04-09

Kazuhiro Ishige , Kazushige Nakagawa , Paolo Salani

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Starshapedeness for fully-nonlinear equations in Carnot groups

2018-04-02

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 natural … 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.

Stability for a strengthened Borell–Brascamp–Lieb inequality

2018-03-21

A.P. Rossi , Paolo Salani

Some overdetermined problems related to the anisotropic capacity

2018-03-13

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 $p$-capacity of a convex set $\Omega \subset \mathbb{R}^N$, with $1<p<N$. In … 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 $\Omega \subset \mathbb{R}^N$, with $1<p<N$. In particular we show that if the Finsler $p$-capacitary potential $u$ associated to $\Omega$ has two homothetic level sets then $\Omega$ is Wulff shape. Moreover, we show that the concavity exponent of $u$ is $q=-(p-1)/(N-p)$ if and only if $\Omega$ is Wulff shape.

Some overdetermined problems related to the anisotropic capacity

2018-01-01

Chiara Bianchini , Giulio Ciraolo , Paolo Salani

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Starshapedeness for fully-nonlinear equations in Carnot groups

2018-01-01

Federica Dragoni , Nicola Garofalo , Paolo Salani

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Optimal Concavity for Newtonian Potentials

2017-12-01

Paolo Salani

In this note I give a short overview about convexity properties of solutions to elliptic equations in convex domains and convex rings and show a result about the optimal concavity … In this note I give a short overview about convexity properties of solutions to elliptic equations in convex domains and convex rings and show a result about the optimal concavity of the Newtonian potential of a bounded convex domain in ℝ n , n ≥ 3, namely: if the Newtonian potential of a bounded domain is ”sufficiently concave”, then the domain is necessarily a ball. This result can be considered an unconventional overdetermined problem. This paper is based on a talk given by the author in Bologna at the ”Bruno Pini Mathematical Analysis Seminar”, which in turn was based on the paper P. Salani, A characterization of balls through optimal concavity for potential functions, Proc. AMS 143 (1) (2015), 173-183.

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Stability for Borell-Brascamp-Lieb Inequalities

2017-01-01

A.P. Rossi , Paolo Salani

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

2017-01-01

Sven Jarohs , Tadeusz Kulczycki , Paolo Salani

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 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 $u$.

Optimal concavity of the torsion function

2017-01-01

Antoine Henrot , Carlo Nitsch , Paolo Salani +1 more

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

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An overdetermined problem for the anisotropic capacity

2016-06-25

Chiara Bianchini , Giulio Ciraolo , Paolo Salani

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Geometric Properties for Parabolic and Elliptic PDE's

2016-01-01

Filippo Gazzola , Kazuhiro Ishige , Carlo Nitsch +1 more

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Stability for Borell-Brascamp-Lieb inequalities

2016-01-01

Alessandro Rossi , Paolo Salani

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 … 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.

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Parabolic Minkowski convolutions of solutions to parabolic boundary value problems

2015-11-20

Kazuhiro Ishige , Paolo Salani

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An overdetermined problem for the anisotropic capacity

2015-09-25

Chiara Bianchini , Giulio Ciraolo , Paolo Salani

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 … 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 $H_0$).

Power concavity in weakly coupled elliptic and parabolic systems

2015-09-12

Kazuhiro Ishige , Kazushige Nakagawa , Paolo Salani

The Hadamard variational formula and the Minkowski problem for p-capacity

2015-09-11

Andrea Colesanti , Kaj Nyström , Paolo Salani +3 more

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Quantitative Borell-Brascamp-Lieb inequalities for compactly supported power concave functions (and some applications)

2015-02-10

Daria Ghilli , Paolo Salani

We strengthen, in two different ways, the so called Borell-Brascamp- Lieb inequality in the class of power concave functions with compact support. As examples of applications we obtain two quantitative … We strengthen, in two different ways, the so called Borell-Brascamp- Lieb inequality in the class of power concave functions with compact support. As examples of applications we obtain two quantitative versions of the Brunn- Minkowski inequality and of the Urysohn inequality for torsional rigidity.

Coauthor	Papers Together
Kazuhiro Ishige	24
Asuka Takatsu	10
Chiara Bianchini	9
Andrea Colesanti	9
Carlo Nitsch	7
Sven Jarohs	7
Tadeusz Kulczycki	7
Giulio Ciraolo	6
Cristina Trombetti	6
Nicola Garofalo	6
Barbara Brandolini	4
Federica Dragoni	4
Andrea Cianchi	3
Lorenzo Brasco	3
Daria Ghilli	3
Antoine Henrot	3
Rolando Magnanini	3
Xi‐Nan Ma	2
Chuanqiang Chen	2
Elisa Francini	2
Paola Cuoghi	2
Kazushige Nakagawa	2
Chiara Bianchini	2
Qing Liu	2
Marco Longinetti	2
A.P. Rossi	2
G. Zhang	1
Stefano Campi	1
Da Yang	1
Jie Xiao	1
Filippo Gazzola	1
Paolo Gronchi	1
Galyna V. Livshyts	1
Kaj Nyström	1
Gianmarco Giovannardi	1
Tatsuki Kawakami	1
Vincenzo Ferone	1
Qing Liu	1
Alessandro Rossi	1
Lei Qin	1
Futoshi Takahashi	1
Massimiliano Bianchini	1