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• W2964049380 abstract "In this paper, by extending the notions of harmonic transplantation and harmonic radius in the Heisenberg group, we give an upper bound for the first eigenvalue for the following Dirichlet problem: $$begin{aligned} (P_{Omega }) left{ begin{array}{rllll} -Delta _{mathbb {H}^1} u &{} = &{} lambda u &{}quad text{ in } &{} Omega u &{} = &{} 0 &{}quad text{ on } &{} partial Omega , end{array} right. end{aligned}$$where ( Omega ) is a regular bounded domain of ( mathbb {H}^1) and (Delta _{mathbb {H}^1}) is the Kohn–Laplace operator. Using a result of Pansu which gives a relation between the volume of (Omega ) and the perimeter of its boundary, we prove that $$begin{aligned} lambda _{1}( Omega ) le C_{Omega } frac{ l_{11}^2 }{ max _{ xi in Omega } r_{Omega }^2(xi )} end{aligned}$$where (l_{11} ) is the first strictly positive zero of the Bessel function of first kind and order 1, ( C_{Omega } ) is a constant depending of ( Omega ), and (r_{Omega }(xi ) ) is the harmonic radius of ( Omega ) at a point (xi ) of (Omega .)" @default.
• W2964049380 created "2019-07-30" @default.
• W2964049380 creator A5023629338 @default.
• W2964049380 creator A5085708894 @default.
• W2964049380 date "2017-03-03" @default.
• W2964049380 modified "2023-09-26" @default.
• W2964049380 title "The First Eigenvalue of the Kohn–Laplace Operator in the Heisenberg Group" @default.
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• W2964049380 doi "https://doi.org/10.1007/s00009-017-0851-8" @default.
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