Inequalities between Dirichlet and Neumann eigenvalues in large dimensions

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• Nikolai Filonov (St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences)
• Wednesday 18 March 2026, 14:00-15:00
• Seminar Room 2, Newton Institute.

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GST - Geometric spectral theory and applications

Let $\Omega\subset\mathbb{R}^{$ be a bounded domain. Denote by $\{\lambda_{k}\}}{\infty}$ (resp. $\{\mu{k}\}_{k=1}^{)$ the eigenvalues of the Laplace operator in $\Omega$ with Dirichlet (resp. Neumann) boundary conditions. Introduce notations $\Phi(d,k,\Omega)=\#\{j:\mu_{j}(\Omega)\le\lambda_{k}(\Omega)\}$, $\Psi(d,k,\Omega)=\Phi(d,k,\Omega)-k$ , so the inequality $\mu_{k+\Psi(d,k,\Omega)}\le\lambda_{k}$ holds true. In 1986, Levine and Weinberger proved the estimate $\Psi(d,k,\Omega)\ge d$ for all convex domains. We show that for $d\gg1$ this result can be improved: $\Psi(d,k,\Omega)\ge C(\frac{e}{2})}{d}$ also for all convex domains. The similar estimate holds for $k=1$: $\Psi(d,1,\Omega)\ge C(\frac{e}{2})^{d}$ for arbitrary domains.

This talk is part of the Isaac Newton Institute Seminar Series series.

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