A Degenerate Elliptic System Solvable by Transport: A Cautionary Example
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915801719635968 |
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| author | Alayón-Solarz, Daniel |
| author_facet | Alayón-Solarz, Daniel |
| contents | We exhibit a one-parameter family of first-order real elliptic systems on the plane whose ellipticity constant degenerates to zero as $δ\to 0$, with condition number $κ= O(δ^{-2})$. For any fixed elliptic solver operating at finite precision, the parameter $δ$ can be chosen small enough to defeat the solver; no uniform numerical scheme based on the ellipticity constant alone can handle the entire family. Despite this, every member of the family is explicitly solvable -- and its initial value problem well posed -- by elementary means once a transport-theoretic invariant is identified. The cost of the transport solution is independent of $δ$. The example serves as a cautionary tale: the ellipticity constant alone does not determine the practical difficulty of a first-order PDE. Before invoking an elliptic solver, one should compute the transport obstruction $G$; its vanishing -- or smallness -- signals structure that standard elliptic methods miss entirely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15479 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Degenerate Elliptic System Solvable by Transport: A Cautionary Example Alayón-Solarz, Daniel Analysis of PDEs Numerical Analysis Complex Variables 35J46 (Primary) 30C62, 65N99 (Secondary) We exhibit a one-parameter family of first-order real elliptic systems on the plane whose ellipticity constant degenerates to zero as $δ\to 0$, with condition number $κ= O(δ^{-2})$. For any fixed elliptic solver operating at finite precision, the parameter $δ$ can be chosen small enough to defeat the solver; no uniform numerical scheme based on the ellipticity constant alone can handle the entire family. Despite this, every member of the family is explicitly solvable -- and its initial value problem well posed -- by elementary means once a transport-theoretic invariant is identified. The cost of the transport solution is independent of $δ$. The example serves as a cautionary tale: the ellipticity constant alone does not determine the practical difficulty of a first-order PDE. Before invoking an elliptic solver, one should compute the transport obstruction $G$; its vanishing -- or smallness -- signals structure that standard elliptic methods miss entirely. |
| title | A Degenerate Elliptic System Solvable by Transport: A Cautionary Example |
| topic | Analysis of PDEs Numerical Analysis Complex Variables 35J46 (Primary) 30C62, 65N99 (Secondary) |
| url | https://arxiv.org/abs/2602.15479 |