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• W3127283689 abstract "Let $mu_2(Omega)$ be the first positive eigenvalue of the Neumann Laplacian in a bounded domain $Omega subset mathbb{R}^N$. It was proved by Szego for $N=2$ and by Weinberger for $N geq 2$ that among all equimeasurable domains $mu_2(Omega)$ attains its global maximum if $Omega$ is a ball. In the present work, we develop the approach of Weinberger in two directions. Firstly, we refine the Szego-Weinberger result for a class of domains of the form $Omega_{text{out}} setminus overline{Omega}_{text{in}}$ which are either centrally symmetric or symmetric of order $2$ (with respect to any coordinate plane $(x_i,x_j)$) by showing that $mu_{2}(Omega_{text{out}} setminus overline{Omega}_{text{in}}) leq mu_2(B_beta setminus overline{B}_alpha)$, where $B_alpha, B_beta$ are balls centered at the origin such that $B_alpha subset Omega_{text{in}}$ and $|Omega_{text{out}}setminus overline{Omega}_{text{in}}|=|B_beta setminus overline{B}_alpha|$. Secondly, we provide Szegő-Weinberger type inequalities for higher eigenvalues by imposing additional symmetry assumptions on the domain. Namely, if $Omega_{text{out}} setminus overline{Omega}_{text{in}}$ is symmetric of order $4$, then we prove $mu_{i}(Omega_{text{out}} setminus overline{Omega}_{text{in}}) leq mu_i(B_beta setminus overline{B}_alpha)$ for $i=3,dots,N+2$, where we also allow $Omega_{text{in}}$ and $B_alpha$ 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 $mu_{N+2}(B_beta setminus overline{B}_alpha)$." @default.
• W3127283689 created "2021-02-15" @default.
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• W3127283689 date "2021-02-11" @default.
• W3127283689 modified "2023-09-27" @default.
• W3127283689 title "Szego-Weinberger type inequalities for symmetric domains with holes" @default.
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