0000000001219025
AUTHOR
F. Chiacchio
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.
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.