0000000000049968
AUTHOR
B. Brandolini
A sharp lower bound for some neumann eigenvalues of the hermite operator
This paper deals with the Neumann eigenvalue problem for the Hermite operator defined in a convex, possibly unbounded, planar domain $\Omega$, having one axis of symmetry passing through the origin. We prove a sharp lower bound for the first eigenvalue $\mu_1^{odd}(\Omega)$ with an associated eigenfunction odd with respect to the axis of symmetry. Such an estimate involves the first eigenvalue of the corresponding one-dimensional problem. As an immediate consequence, in the class of domains for which $\mu_1(\Omega)=\mu_1^{odd}(\Omega)$, we get an explicit lower bound for the difference between $\mu(\Omega)$ and the first Neumann eigenvalue of any strip.
Comparison results for a linear elliptic equation with mixed boundary conditions
In this paper we study a linear elliptic equation having mixed boundary conditions, defined in a connected open set $\Omega $ of $\mathbb{R}^{n}$. We prove a comparison result with a suitable ``symmetrized'' Dirichlet problem which cannot be uniformly elliptic depending on the regularity of $ \partial \Omega $. Regularity results for non-uniformly elliptic equations are also given.
Some remarks on the extinction time for the mean curvature flow
We write some consideratons on the extinction time for the mean curvature flow
On a time-depending Monge-Ampère type equation
Abstract In this paper, we prove a comparison result between a solution u ( x , t ) , x ∈ Ω ⊂ R 2 , t ∈ ( 0 , T ) , of a time depending equation involving the Monge–Ampere operator in the plane and the solution of a conveniently symmetrized parabolic equation. To this aim, we prove a derivation formula for the integral of a smooth function g ( x , t ) over sublevel sets of u , { x ∈ Ω : u ( x , t ) ϑ } , ϑ ∈ R , having the same perimeter in R 2 .
Some remarks on nonlinear elliptic problems involving Hardy potentials
In this note we prove an Hardy type inequality with a remainder term, where the potential depends only on a group of variables. Such a result allows us to show the existence of entropy solutions to a class of elliptic P.D.E.'s.