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SUMMARY:Inequalities between Dirichlet and Neumann eigenvalues in large di
 mensions - Nikolai Filonov (St. Petersburg Department of Steklov Mathemati
 cal Institute of Russian Academy of Sciences)
DTSTART:20260318T140000Z
DTEND:20260318T150000Z
UID:TALK242929@talks.cam.ac.uk
DESCRIPTION:Let $\\Omega\\subset\\mathbb{R}^{d}$ be a bounded domain. Deno
 te by $\\{\\lambda_{k}\\}_{k=1}^{\\infty}$ (resp. $\\{\\mu_{k}\\}_{k=1}^{\
 \infty})$ the eigenvalues of the Laplace operator in $\\Omega$ with Dirich
 let (resp. Neumann) boundary conditions. Introduce notations\n$\\Phi(d\,k\
 ,\\Omega)=\\#\\{j:\\mu_{j}(\\Omega)\\le\\lambda_{k}(\\Omega)\\}$\, $\\Psi(
 d\,k\,\\Omega)=\\Phi(d\,k\,\\Omega)-k$ \,\nso the inequality\n$\\mu_{k+\\P
 si(d\,k\,\\Omega)}\\le\\lambda_{k}$\nholds true. In 1986\, Levine and Wein
 berger proved the estimate $\\Psi(d\,k\,\\Omega)\\ge d$ for all convex dom
 ains. We show that for $d\\gg1$ this result can be improved:\n$\\Psi(d\,k\
 ,\\Omega)\\ge C(\\frac{e}{2})^{d}$\nalso for all convex domains. The simil
 ar estimate holds for $k=1$:\n$\\Psi(d\,1\,\\Omega)\\ge C(\\frac{e}{2})^{d
 }$\nfor arbitrary domains.
LOCATION:Seminar Room 2\, Newton Institute
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