Quadratic cones on which few harmonic functions vanish
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909248570523648 |
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| author | Greilhuber, Josef Eberhard |
| author_facet | Greilhuber, Josef Eberhard |
| contents | We show that, in dimension three and higher, the space of harmonic functions vanishing on the cone defined by a generically chosen harmonic quadratic polynomial is two-dimensional. This phenomenon is surprisingly robust, generalizing to arbitrary elliptic differential operators of second order, with the cone replaced by the level set of a solution at a nondegenerate critical value. As long as the tangent cone to the level set at the critical point satisfies a certain genericity condition, the space of solutions vanishing on the level set is at most two-dimensional. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07039 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quadratic cones on which few harmonic functions vanish Greilhuber, Josef Eberhard Analysis of PDEs Classical Analysis and ODEs Primary: 31B05, Secondary: 35J05, 35J15, 33C55, 33C50 We show that, in dimension three and higher, the space of harmonic functions vanishing on the cone defined by a generically chosen harmonic quadratic polynomial is two-dimensional. This phenomenon is surprisingly robust, generalizing to arbitrary elliptic differential operators of second order, with the cone replaced by the level set of a solution at a nondegenerate critical value. As long as the tangent cone to the level set at the critical point satisfies a certain genericity condition, the space of solutions vanishing on the level set is at most two-dimensional. |
| title | Quadratic cones on which few harmonic functions vanish |
| topic | Analysis of PDEs Classical Analysis and ODEs Primary: 31B05, Secondary: 35J05, 35J15, 33C55, 33C50 |
| url | https://arxiv.org/abs/2407.07039 |