A Degenerate Elliptic System Solvable by Transport: A Cautionary Example

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.

Source: arXiv:2602.15479v1 - http://arxiv.org/abs/2602.15479v1 PDF: https://arxiv.org/pdf/2602.15479v1 Original Link: http://arxiv.org/abs/2602.15479v1