Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.17721 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912972777979904 |
|---|---|
| author | Lopez-Gomez, J. Sahuquillo, A. |
| author_facet | Lopez-Gomez, J. Sahuquillo, A. |
| contents | 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 $\varrho>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17721 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Limiting behavior of principal eigenvalues for a class of mixed boundary value problems as the measure of the support domain goes to zero Lopez-Gomez, J. Sahuquillo, A. Analysis of PDEs 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 $\varrho>0$. |
| title | Limiting behavior of principal eigenvalues for a class of mixed boundary value problems as the measure of the support domain goes to zero |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.17721 |