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 $\delta \to 0$, with condition number $\kappa =O({\delta }^{-2})$. For any fixed elliptic solver operating at finite precision, the parameter $\delta$ 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 $\delta$. 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.