The hot spots conjecture on Gaussian spaces
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913063438909440 |
|---|---|
| author | Hua, Bobo Sun, Jin |
| author_facet | Hua, Bobo Sun, Jin |
| contents | We study the hot spots conjecture for domains in the Gaussian space $(\mathbb{R}^n, (2π)^{-n/2} e^{-|x|^2/2} dx)$ for $n \ge 2$. Given a bounded domain $Ω$ with a piecewise smooth boundary, we consider the first nontrivial eigenfunction of the Ornstein--Uhlenbeck operator $L_γ= Δ- \langle x, \nabla \rangle$ subject to Neumann or mixed Dirichlet--Neumann boundary conditions, and prove that its extrema are attained only on the boundary $\partialΩ$.
More precisely, we establish the conjecture for two classes of domains: (i) lip domains in Gaussian spaces with mixed boundary conditions, and (ii) $n$-symmetric domains whose intersection with some orthant is a lip domain. As a corollary, we show that any first nontrivial Neumann eigenfunction of a $2$-symmetric domain in the two-dimensional Gaussian space has no interior extrema, provided the second Neumann eigenvalue is simple. Our approach is based on a variational principle for the Hodge Laplacian on weighted manifolds and the Hodge decomposition of differential $1$-forms on Lipschitz domains, extending the variational method of Kennedy--Rohleder from the Euclidean setting to Gaussian spaces. Although de Dios Pont has shown that the hot spots conjecture can fail for certain convex domains endowed with suitable log-concave measures, our results identify broad classes of domains for which the conjecture remains valid in Gaussian spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20663 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The hot spots conjecture on Gaussian spaces Hua, Bobo Sun, Jin Spectral Theory Differential Geometry We study the hot spots conjecture for domains in the Gaussian space $(\mathbb{R}^n, (2π)^{-n/2} e^{-|x|^2/2} dx)$ for $n \ge 2$. Given a bounded domain $Ω$ with a piecewise smooth boundary, we consider the first nontrivial eigenfunction of the Ornstein--Uhlenbeck operator $L_γ= Δ- \langle x, \nabla \rangle$ subject to Neumann or mixed Dirichlet--Neumann boundary conditions, and prove that its extrema are attained only on the boundary $\partialΩ$. More precisely, we establish the conjecture for two classes of domains: (i) lip domains in Gaussian spaces with mixed boundary conditions, and (ii) $n$-symmetric domains whose intersection with some orthant is a lip domain. As a corollary, we show that any first nontrivial Neumann eigenfunction of a $2$-symmetric domain in the two-dimensional Gaussian space has no interior extrema, provided the second Neumann eigenvalue is simple. Our approach is based on a variational principle for the Hodge Laplacian on weighted manifolds and the Hodge decomposition of differential $1$-forms on Lipschitz domains, extending the variational method of Kennedy--Rohleder from the Euclidean setting to Gaussian spaces. Although de Dios Pont has shown that the hot spots conjecture can fail for certain convex domains endowed with suitable log-concave measures, our results identify broad classes of domains for which the conjecture remains valid in Gaussian spaces. |
| title | The hot spots conjecture on Gaussian spaces |
| topic | Spectral Theory Differential Geometry |
| url | https://arxiv.org/abs/2412.20663 |