Plane-wave representation for the Laplace--Beltrami equation on a sphere. Application to the Green's function
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866908913731895296 |
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| author | Shanin, Andrey V. Kunz, Valentin D. Assier, Raphael C. |
| author_facet | Shanin, Andrey V. Kunz, Valentin D. Assier, Raphael C. |
| contents | We propose an extension of the plane-wave representation for wave fields defined on the real sphere $\mathcal{S}^2$. This representation is well-known in the planar setting but has never been developed for curved surfaces. To achieve this, we need to carefully study the geometry of the complexification of $\mathcal{S}^2$ and the properties of the Laplace--Beltrami operator, while using concepts of multidimensional complex analysis. We extend the region of validity of such plane-wave representation by developing a sliding-contours method. Our methodology is illustrated through the study of the Green's function on the real sphere. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_24417 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Plane-wave representation for the Laplace--Beltrami equation on a sphere. Application to the Green's function Shanin, Andrey V. Kunz, Valentin D. Assier, Raphael C. Analysis of PDEs We propose an extension of the plane-wave representation for wave fields defined on the real sphere $\mathcal{S}^2$. This representation is well-known in the planar setting but has never been developed for curved surfaces. To achieve this, we need to carefully study the geometry of the complexification of $\mathcal{S}^2$ and the properties of the Laplace--Beltrami operator, while using concepts of multidimensional complex analysis. We extend the region of validity of such plane-wave representation by developing a sliding-contours method. Our methodology is illustrated through the study of the Green's function on the real sphere. |
| title | Plane-wave representation for the Laplace--Beltrami equation on a sphere. Application to the Green's function |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.24417 |