A Dynamical Approach to the Berezin-Li-Yau Inequality
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| Format: | Preprint |
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2025
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| _version_ | 1866917527442948096 |
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| author | Alexa, Anton |
| author_facet | Alexa, Anton |
| contents | We develop a dynamical method for proving the sharp Berezin-Li-Yau inequality. The approach is based on the volume-preserving mean curvature flow and a new monotonicity principle for the Riesz mean $R_Λ(Ω_t)$. For convex domains we show that $R_Λ$ is monotone non-decreasing along the flow. The key input is a geometric correlation inequality between the boundary spectral density $Q_Λ$ and the mean curvature $H$, established in all dimensions: in $d=2$ via a near-disk Fourier analysis, and in $d\ge 3$ via the boundary Weyl expansion together with a local spectral rigidity argument near the ball, with a first-zero exclusion principle closing the global step. Since the flow converges smoothly to the ball, the monotonicity implies the sharp Berezin-Li-Yau bound for every smooth convex domain. As an application, we obtain a sharp dynamical Cesàro-Pólya inequality for eigenvalue averages. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_08966 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Dynamical Approach to the Berezin-Li-Yau Inequality Alexa, Anton Differential Geometry Spectral Theory 35P15, 58J50, 53C44, We develop a dynamical method for proving the sharp Berezin-Li-Yau inequality. The approach is based on the volume-preserving mean curvature flow and a new monotonicity principle for the Riesz mean $R_Λ(Ω_t)$. For convex domains we show that $R_Λ$ is monotone non-decreasing along the flow. The key input is a geometric correlation inequality between the boundary spectral density $Q_Λ$ and the mean curvature $H$, established in all dimensions: in $d=2$ via a near-disk Fourier analysis, and in $d\ge 3$ via the boundary Weyl expansion together with a local spectral rigidity argument near the ball, with a first-zero exclusion principle closing the global step. Since the flow converges smoothly to the ball, the monotonicity implies the sharp Berezin-Li-Yau bound for every smooth convex domain. As an application, we obtain a sharp dynamical Cesàro-Pólya inequality for eigenvalue averages. |
| title | A Dynamical Approach to the Berezin-Li-Yau Inequality |
| topic | Differential Geometry Spectral Theory 35P15, 58J50, 53C44, |
| url | https://arxiv.org/abs/2512.08966 |