Szegő-Weinberger type inequalities for symmetric domains with holes

Abstract

Let 𝜇2(Ω) be the first positive eigenvalue of the Neumann Laplacian in a bounded domain Ω ⊂ R 𝑁 . It was proved by Szegő for 𝑁 = 2 and by Weinberger for 𝑁 ≥ 2 that among all equimeasurable domains 𝜇2(Ω) attains its global maximum if Ω is a ball. In the present work, we develop the approach of Weinberger in two directions. Firstly, we refine the Szegő-Weinberger result for a class of domains of the form Ωout ∖ Ωin which are either centrally symmetric or symmetric of order 2 (with respect to any coordinate plane (𝑥𝑖, 𝑥𝑗)) by showing that 𝜇2(Ωout ∖ Ωin) ≤ 𝜇2(𝐵 𝛽 ∖ 𝐵𝛼), where 𝐵𝛼, 𝐵 𝛽 are balls centered at the origin such that 𝐵𝛼 ⊂ Ωin and |Ωout ∖ Ωin| = |𝐵 𝛽 ∖ 𝐵𝛼|. Secondly, we provide Szegő-Weinberger type inequalities for higher eigenvalues by imposing additional symmetry assumptions on the domain. Namely, if Ωout ∖ Ωin is symmetric of order 4, then we prove 𝜇𝑖(Ωout ∖ Ωin) ≤ 𝜇𝑖(𝐵 𝛽 ∖ 𝐵𝛼) for 𝑖 = 3, . . . , 𝑁 + 2, where we also allow Ωin and 𝐵𝛼 to be empty. If 𝑁 = 2 and the domain is symmetric of order 8, then the latter inequality persists for 𝑖 = 5. Counterexamples to the obtained inequalities for domains outside of the considered symmetry classes are given. The existence and properties of nonradial domains with required symmetries in higher dimensions are discussed. As an auxiliary result, we obtain the non-radiality of the eigenfunctions associated to 𝜇𝑁+2(𝐵 𝛽 ∖ 𝐵𝛼).

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“…However, the strict monotonicity has been obtained by Anoop, Bobkov, and Sasi in [8]. The results related to the optimization of eigenvalues with respect to other boundary conditions can be found in [5][6][7]. For further literature and open problems in this direction, we refer to the book [23]; see also [22] for recent developments in spectral geometry.…”

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“…Apart from the Dirichlet eigenvalue problem on A s , one may also consider Neumann boundary conditions [2,68], or mixed boundary conditions [40,58], or the Steklov eigenvalue problem [31,55], or the mixed Steklov-Dirichlet problem [41,68], or the p-Laplacian [1,17], and so on.…”