| Abstract: |
| In this paper we characterize the limiting behavior of the principal eigenvalue, $\sigma_1[-\Delta,\beta,\Omega]$, of the boundary value problem (1.1) as the Lebesgue measure of the underlying domain, $\Omega$, tends to zero. Naturally, the domains $\Omega$ are assumed to be included on a fixed open set $D$ such that $\beta\in\mathcal{C}(D)$, and they satisfy $\bar\Omega\subset D$. Our main result establishes that, in the classical case when $\inf_{D}\beta >0$,
$$
\lim_{|\Omega|\downarrow 0}\sigma_1[-\Delta,\beta,\Omega] =+\infty,
$$
whereas
$$
\lim_{|\Omega|\downarrow 0}\sigma_1[-\Delta,\beta,\Omega] =-\infty\;\;\hbox{if}\;\; \sup_{D}\beta < 0,
$$
which is a surprising result at the light of the classical existing theorems for Dirichlet boundary conditions. Furthermore, in the special case when $\beta>0$ is a constant, we can prove that
$$
\lim_{R\downarrow 0}\left( R \sigma_1[-\Delta,\beta,B_R]\right)=\beta \frac{\mathrm{Area}(\partial B_1)}{|B_1|},
$$
where, we are denoting $B_\varrho:=\{x \in \mathbb{R}^N \;:\;|x|< \varrho \}$ for all $\varrho>0$. This is a joint work with Juli\`{a}n L\`{o}pez-G\`{o}mez. |
|