Paolo Salani

Brunn-Minkowski inequalities for two functionals involving the p -Laplace operator

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.

Starshapedness of level sets of solutions to elliptic PDEs

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.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.

Convexity breaking of the free boundary for porous medium equations

Is then Pφ(t) convex for every t > 0? (See Section 2 for the definition of α-concavity.) The porous medium equation provides a simple model in many physical situations, in particular, the flow of an isentropic gas through a porous medium; in such a case, u and um−1 represent the density and the pressure of the gas, respectively. Due to its practical interest, regularity and geometric properties of the free boundary ∂Pφ(t) have been extensively studied by many

A characterization of balls through optimal concavity for potential functions

Let

Power Concavity for Solutions of Nonlinear Elliptic Problems in Convex Domains

p\in(1,n)

An overdetermined problem with non-constant boundary condition

. If

Stability for Borell-Brascamp-Lieb Inequalities

\Omega

Optimal concavity of the torsion function

is a convex domain in

Spatial concavity of solutions to parabolic systems

\rn

Stability for a strengthened Borell–Brascamp–Lieb inequality

whose