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

In this paper we characterize the limiting behavior of the principal eigenvalue, ${\text{s}}_1[-\text{D},\text{b},\text{Ø}]$, of the boundary value problem \eqref{1.1} as the Lebesgue measure of the underlying domain, $\text{Ø}$, tends to zero. Naturally, the domains $\text{Ø}$ are assumed to be included on a fixed open set $D$ such that $\text{b}\in \text{mc}C(D)$, and they satisfy $\overline{\text{Ø}}\subset D$. Our main result establishes that, in the classical case when ${\inf }_D\text{b}>0$, $$\underset{\mathrm{\mid }\text{Ø}\mathrm{\mid }\text{da}0}{\lim }{\text{s}}_1[-\text{D},\text{b},\text{Ø}]=+\mathrm{\infty },$$ whereas $$\underset{\mathrm{\mid }\text{Ø}\mathrm{\mid }\text{da}0}{\lim }{\text{s}}_1[-\text{D},\text{b},\text{Ø}]=-\mathrm{\infty }\text{if}\underset{D}{\sup }\text{b}<0,$$ which is a surprising result at the light of the classical existing theorems for Dirichlet boundary conditions. Furthermore, in the special case when $\text{b}>0$ is a constant, we can prove that $$\underset{R\text{da}0}{\lim }(R{\text{s}}_1[-\text{D},\text{b},B_R])=\text{b}\frac{\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}(\text{p}{B}_1)}{\mathrm{\mid }{B}_1\mathrm{\mid }},$$ where, we are denoting $B_{\varrho }:=\{x\in {\mathbb{R}}^N:\mathrm{\mid }x\mathrm{\mid }<\varrho \}$ for all $\varrho >0$.