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

J. Lopez-Gomez, A. Sahuquillo

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 $\barØ\subset D$. 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 $B_\varrho:=\{x\in\R^N\;:\;|x|<\varrho\}$ for all $\varrho>0$.

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, , of the boundary value problem \eqref{1.1} as the Lebesgue measure of the underlying domain, , tends to zero. Naturally, the domains are assumed to be included on a fixed open set such that , and they satisfy . Our main result establishes that, in the classical case when , whereas which is a surprising result at the light of the classical existing theorems for Dirichlet boundary conditions. Furthermore, in the special case when is a constant, we can prove that where, we are denoting for all .
Paper Structure (13 sections, 6 theorems, 174 equations, 3 figures)

This paper contains 13 sections, 6 theorems, 174 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['th1.1']}
  4. Case 1: $\mathcal{B}_0, \mathcal{B}_L \in \{ \mathcal{D}, \mathcal{N} \}$
  5. Case 2: $\mathcal{B}_0 = \mathcal{R}_\beta$, with $\beta_0 \in\mathbb{R}\setminus\{0\}$, and $\mathcal{B}_L = \mathcal{D}$
  6. Case 3: $\mathcal{B}_0 = \mathcal{D}$ and $\mathcal{B}_L = \mathcal{R}_\beta$, with $\beta_\omega \in \mathbb{R} \setminus \{0\}$
  7. Case 4: $\mathcal{B}_0 = \mathcal{R}_\beta$, with $\beta_0 \in \mathbb{R} \setminus \{0\}$, and $\mathcal{B}_L = \mathcal{N}$
  8. Case 5: $\mathcal{B}_0 = \mathcal{N}$ and $\mathcal{B}_L = \mathcal{R}_\beta$, with $\beta_\omega \in \mathbb{R} \setminus \{0\}$
  9. Case 6: $\mathcal{B}_0 = \mathcal{B}_L = \mathcal{R}_\beta$, with $\beta_0,\beta_\omega \in \mathbb{R} \setminus \{0\}$
  10. Proof of Theorem \ref{['th1.2']}
  11. Proof of (\ref{['1.4']})
  12. Proof of (\ref{['1.5']})
  13. Asymptotic expansion of $\sigma_1[-\Delta,\beta_\ell,B_R]$ as $R\downarrow 0$ in case $\beta_\ell>0$

Key Result

Theorem 1.1

The following conditions are satisfied.

Figures (3)

  • Figure 1: The graphs of the line $-\frac{s}{L \beta_0}$, in black, and $\tan s$, in blue.
  • Figure 2: The graphs of the hyperbola $\tfrac{\beta_0}{s}$, in black, and the curve $\tan (sL)$ for $L_1=1$, in blue, and for $L_2=0.2$, in red.
  • Figure 3: The plots of the hyperbola $-\tfrac{\beta_0}{s}$, in black, and $\tanh(sL)$ for $L_1=1$, in blue, and $L_2=0.2$, in red.

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 5.1
  • proof