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

TL;DR

This paper investigates how the principal eigenvalue of a boundary value problem behaves as the domain's measure shrinks to zero, revealing divergent limits depending on the sign of the coefficient function.

Contribution

It provides a detailed characterization of the eigenvalue's limiting behavior for small domains, including surprising divergence results and explicit limits in special cases.

Findings

Eigenvalue tends to +infinity if the coefficient is positive.

Eigenvalue tends to -infinity if the coefficient is negative.

Explicit limit formula for constant positive coefficient in small balls.

Abstract

In this paper we characterize the limiting behavior of the principal eigenvalue, $\s_1[-\D,\b,\O]$, of the boundary value problem \eqref{1.1} as the Lebesgue measure of the underlying domain, $\O$, tends to zero. Naturally, the domains $\O$ are assumed to be included on a fixed open set $D$ such that $\b\in\mc{C}(D)$, and they satisfy $\bar\O\subset D$. Our main result establishes that, in the classical case when $\inf_{D}\b >0$, $$ \lim_{|\O|\da 0}\s_1[-\D,\b,\O] =+\infty, $$ whereas $$ \lim_{|\O|\da 0}\s_1[-\D,\b,\O] =-\infty\;\;\hbox{if}\;\; \sup_{D}\b <0, $$ 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 $B_\varrho:=\{x\in\R^N\;:\;|x|<\varrho\}$ for all $\varrho>0$.