Arrieta, Jose M.
Elliptic Equations, Principal Eigenvalue and
Dependence on the Domain.
(69K, LaTeX)
ABSTRACT. We consider a general second order uniformly elliptic
differential operator $L$ and also the set $\Theta$ of
all open sets (not neccessarily smooth) in the unit
ball of $\R^n$. We define a metric $d$ in this set
(up to an equivalence relation $\sim$) that makes the space
$(\Theta/\sim, d)$ a complete metric space. We show that
the principal eigenvalue and eigenfunction of $L$ are
continuous with the metric $d$. Similar results are
obtained for the solutions of the equation $Lv=f$.