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ON THE LOWEST EIGENVALUE OF THE FRACTIONAL LAPLACIAN FOR THE INTERSECTION OF TWO DOMAINS

Part of: Spectral theory and eigenvalue problems Partial differential operators

Published online by Cambridge University Press:  02 December 2022

ANH TUAN DUONG Open the ORCID record for ANH TUAN DUONG [Opens in a new window]  and
VAN HOANG NGUYEN Open the ORCID record for VAN HOANG NGUYEN [Opens in a new window]
ANH TUAN DUONG*
Affiliation:
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hai Ba Trung, Ha Noi, Vietnam
VAN HOANG NGUYEN
Affiliation:
Department of Mathematics, FPT University, Ha Noi, Vietnam e-mail: vanhoang0610@yahoo.com and hoangnv47@fe.edu.vn
*
e-mail: tuan.duonganh@hust.edu.vn
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Abstract

We extend a result of Lieb [‘On the lowest eigenvalue of the Laplacian for the intersection of two domains’, Invent. Math. 74(3) (1983), 441–448] to the fractional setting. We prove that if A and B are two bounded domains in $\mathbb R^N$ and $\lambda _s(A)$, $\lambda _s(B)$ are the lowest eigenvalues of $(-\Delta )^s$, $0<s<1$, with Dirichlet boundary conditions, there exists some translation $B_x$ of B such that $\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)$. Moreover, without the boundedness assumption on A and B, we show that for any $\varepsilon>0$, there exists some translation $B_x$ of B such that $\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)+\varepsilon .$

Keywords

lowest eigenvaluefractional Laplacianintersection of two domainseigenvalue problem

MSC classification

Primary: 35P15: Estimation of eigenvalues, upper and lower bounds
Secondary: 47F05: Partial differential operators (should also be assigned at least one other classification number in section 47)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 2 , October 2023 , pp. 290 - 297
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

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