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ON THE LOWEST EIGENVALUE OF THE FRACTIONAL LAPLACIAN FOR THE INTERSECTION OF TWO DOMAINS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 108 / Issue 2 / October 2023
- Published online by Cambridge University Press:
- 02 December 2022, pp. 290-297
- Print publication:
- October 2023
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- Article
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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 
$\lambda _s(A\cap B_x)< \lambda _s(A)+\lambda _s(B)+\varepsilon .$
