The hot spots conjecture for some non-convex polygons
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912398459273216 |
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| author | Hatcher, Lawford |
| author_facet | Hatcher, Lawford |
| contents | We give an elementary new proof of the hot spots conjecture for L-shaped domains. This result, in addition to a new eigenvalue inequality, allows us to locate the hot spots in Swiss cross translation surfaces. We then prove, in several cases, that first mixed Dirichlet-Neumann eigenfunctions of the Laplacian on L-shaped domains also have no interior critical points. As a combination of these results, we prove the hot spots conjecture for five classes of domains tiled by L-shaped domains, including a class of non-simply connected domains. An interesting feature of the proofs is that we make positive use of the lack of regularity of eigenfunctions on non-convex polygons. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19508 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The hot spots conjecture for some non-convex polygons Hatcher, Lawford Analysis of PDEs Spectral Theory 35P05, 35B38, 35J05, 35J25, 58J50 We give an elementary new proof of the hot spots conjecture for L-shaped domains. This result, in addition to a new eigenvalue inequality, allows us to locate the hot spots in Swiss cross translation surfaces. We then prove, in several cases, that first mixed Dirichlet-Neumann eigenfunctions of the Laplacian on L-shaped domains also have no interior critical points. As a combination of these results, we prove the hot spots conjecture for five classes of domains tiled by L-shaped domains, including a class of non-simply connected domains. An interesting feature of the proofs is that we make positive use of the lack of regularity of eigenfunctions on non-convex polygons. |
| title | The hot spots conjecture for some non-convex polygons |
| topic | Analysis of PDEs Spectral Theory 35P05, 35B38, 35J05, 35J25, 58J50 |
| url | https://arxiv.org/abs/2405.19508 |