A sharp upper bound for the first Dirichlet eigenvalue of cone-like domains
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Archiv der Mathematik
A sharp upper bound for the first Dirichlet eigenvalue of cone-like domains Abdelhalim Hasnaoui
and Abir Sboui
Abstract. The aim of this paper is to give a complementary upper boundtype isoperimetric inequality for the fundamental Dirichlet eigenvalue of a bounded domain completely contained in a cone. This inequality is a counterpart to the Ratzkin inequality for Euclidean wedge domains in higher dimensions. We also give a new version of the Crooke–Sperb inequality involving a new geometric quantity for the first eigenfunction of the Dirichlet Laplacian for such a class of domains. Mathematics Subject Classification. Primary 35P15, Secondary 58E30. Keywords. Fundamental eigenvalue, Upper bound, Cone-like domain, Crooke–Sperb inequality.
1. Introduction. For the Euclidian bounded domain, the Rayleigh–Faber– Krahn ([2,6]) inequality states that the first Dirichlet eigenvalue is minimized for the ball of the same volume. For a planar star-shaped domain, P´ olya and Szeg¨o [7] offered a counterpart to this inequality which has been recently extended to higher dimensions by Freitas and Krejˇciˇrik [3]. In 1960, Payne and Weinberger [8] proved a Rayleigh–Faber–Krahn type inequality for the case of a bounded domain completely contained in the wedge of angle less than or equal to π. The Payne–Weinberger inequality improved on the classical Faber–Krahn inequality for certain shapes, namely triangles. For such domains, Hasnaoui and Hermi [4] gave a weighted version of the P´ olya and Szeg¨o inequality. Finally, in 2010, Ratzkin [9] generalized the Payne–Weinberger inequality to higher dimensions. Then the natural question is to the find the counterpart to the Ratzkin inequality which also in some cases improves the Freitas–Krejˇciˇrik inequality.
A. Hasnaoui and A. Sboui
Arch. Math.
In this paper, we answer this question. More precisely, by introducing a new geometric factor Bα which will be defined appropriately, we give an upper bound for the first Dirichlet eigenvalue of cone-like domains (bounded domains completely contained in a cone). By applying the method of the present paper, we also give a lower bound for the relative torsional rigidity (see [4] and [5]) and a weighted version of the Crooke–Sperb inquality [1]. 2. Preliminary tools and main result. To state our result, we need to introduce some notations and definitions. Let Ω be a domain in the unit sphere Sn−1 , and let x ∈ Ω} (2.1) W = {x ∈ Rn : |x| be the cone over Ω. For r > 0, we define the cone x ∈ Ω}. Cr = {x ∈ Rn : 0 < |x| < r, |x|
(2.2)
Let μ be the first Dirichlet eigenvalue of the Laplacian on Ω, with positive eigenfunction ρ. We normalize ρ so that ρ2 dσ = 1. Ω
Here dσ denotes the measure on Sn−1 . Observe that 2−n x 2−n 2 α α= + ( ) + μ, w(x) = |x| ρ( ), |x| 2 2
(2.3)
is a positive harmonic and homogeneous function of degree α in W, which is zero on the boundary of W. Let D be the intersection of the cone W and a star-shaped n-dimensional domain with respect to the origin and Γ be the part of the boundary of D contained
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