Limiting behavior of principal eigenvalues for a class of mixed boundary value problems as the measure of the support domain goes to zero

Abstract

In this paper we characterize the limiting behavior of the principal eigenvalue, $\s_1[-\D,\b,Ø]$, of the boundary value problem \eqref{1.1} as the Lebesgue measure of the underlying domain, $Ø$, tends to zero. Naturally, the domains $Ø$ are assumed to be included on a fixed open set $D$ such that $\b\in\mc{C}(D)$, and they satisfy $\overline{Ø}\subset D$. Our main result establishes that, in the classical case when $\inf_{D}\b >0$, $$ \lim_{|Ø|\da 0}\s_1[-\D,\b,Ø] =+\infty, $$ whereas $$ \lim_{|Ø|\da 0}\s_1[-\D,\b,Ø] =-\infty\;\;\hbox{if}\;\; \sup_{D}\b <0, $$ which is a surprising result at the light of the classical existing theorems for Dirichlet boundary conditions. Furthermore, in the special case when $\b>0$ is a constant, we can prove that $$ \lim_{R\da 0}\left( R \s_1[-\D,\b,B_R]\right)=\b \frac{\mathrm{Area}(\p B_1)}{|B_1|}, $$ where, we are denoting $B_{\varrho }:=\{x\in {\mathbb{R}}^N:|x|<\varrho \}$ for all $\varrho >0$.

arXiv comment18 pages, 3 figures