The unresolved cases of Hörmander's Cauchy problem for hyperbolic operators with double characteristics: the equality case of the Ivrii–Petkov condition, spectral transition of the Hamilton map, and tangent bicharacteristics

Hörmander's 1977 paper 'The Cauchy problem for differential equations with double characteristics' proved $C^\infty$ well-posedness of the Cauchy problem for hyperbolic operators with double characteristics under a strict form of the Ivrii–Petkov condition on the subprincipal symbol and constant-type hypotheses on the double characteristic manifold, and it explicitly recorded what its methods could not cover. Subsequent work (Bernardi, Nishitani, Parenti, Parmeggiani, Lascar, Petkov and others, 2008–2026) resolved parts of this program and delineated the open core. Of the six issues the source lists, the effectively hyperbolic case had already been settled by Melrose (1983) following Ivrii. The surviving open core is: (1) the equality case of the Ivrii–Petkov condition when $\operatorname{Tr}^+F_p>0$ on the double set, where the degenerate sub-regime $\operatorname{Tr}^+F_p=0$ (the Levi condition) is characterized only in the absence of null bicharacteristics tangent to $\Sigma$; (2) well-posedness across loci where the spectral type of the Hamilton map changes (transition) and on singular double sets, including necessity of Melrose's conjectured distance condition; (3) the tangent-bicharacteristic regime, where $C^\infty$ well-posedness fails in general and sharp Gevrey thresholds are known only for model classes; together with the source's systems question (kernels of dimension greater than one) and the sharpness of the $m-2$ derivative loss, whose later status is only partially mapped. These questions remain open.

Background

The Cauchy problem for a linear hyperbolic differential operator $P$ of order $m$ asks for solutions of $Pu=f$ with data prescribed on the time hypersurface $x_0=0$; it is well posed (in Hadamard's sense) when solutions exist, are unique, and depend continuously on the data. For strictly hyperbolic operators — those whose principal symbol $p_m$, the top-order homogeneous part of the full symbol viewed as a function on phase space $(x,\xi)$, has $m$ distinct real roots in $\xi_0$ — this is classical (Leray, Gårding). The difficult frontier is the case of double characteristics: points $\rho$ of the double set $\Sigma=\{p_m=0,\ dp_m=0\}$ where the principal symbol vanishes together with all its first derivatives. There the answer depends on the lower-order terms and on the symplectic geometry of $\Sigma$, and a complete characterization has been the goal of a research program running from the 1970s to the present.

Two invariants at a double characteristic govern the theory. The first is the Hamilton map (also called the fundamental matrix) $F_p(\rho)$: the linearization at $\rho$ of the Hamilton vector field $H_p=\sum_j(\partial_{\xi_j}p\,\partial_{x_j}-\partial_{x_j}p\,\partial_{\xi_j})$ of the principal symbol, which induces a well-defined linear map on the tangent space of $\Sigma$ when $\Sigma$ is a smooth manifold on which the Hessian of $p_m$ has constant rank. Write $\operatorname{Tr}^+F_p(\rho)$ for the sum of the positive imaginary parts of the eigenvalues of $F_p(\rho)$. If $F_p(\rho)$ has a pair of real nonzero eigenvalues $\pm e(\rho)$ (the remaining eigenvalues being purely imaginary), the point is called effectively hyperbolic; in that case Melrose (1983), building on Ivrii's earlier work, proved that the Cauchy problem is well posed in $C^\infty$ for arbitrary lower-order terms. Otherwise the spectrum is purely imaginary and the lower-order terms matter. The second invariant is the subprincipal symbol $P_{sub}$, the invariantly defined next-to-leading part of the operator (for a differential operator, the order $m-1$ symbol corrected by $\frac{i}{2}\sum_j\partial^2 p_m/\partial x_j\partial\xi_j$).

Ivrii and Petkov (1974) proved that $C^\infty$ well-posedness forces $P_{sub}$ to be real and to satisfy, at every point of the non-effectively hyperbolic part of $\Sigma$ (where $F_p(\rho)$ has no nonzero real eigenvalue),

$-\operatorname{Tr}^+F_p(\rho)\ \le\ P_{sub}(\rho)\ \le\ \operatorname{Tr}^+F_p(\rho).$

Hörmander (1977) proved the matching sufficiency under structural hypotheses: $\Sigma$ a $C^\infty$ manifold on which the Hessian of $p_m$ has constant rank and constant symplectic type (in his classification of the quadratic homogeneous part into types Ia–Ie and II, with the number of blocks of each type fixed along $\Sigma$), and the subprincipal symbol satisfying a strict version of the inequality above, that is, keeping a uniform margin away from the endpoints $\pm\operatorname{Tr}^+F_p$. That the margin is a proof artifact rather than the true boundary is visible in the constant-coefficient model: an operator with quadratic part $Q$ and constant subprincipal term $S$ is well posed exactly when $Q$ is effectively hyperbolic or $-\sum_j\mu_j\le S\le\sum_j\mu_j$, where $\pm i\mu_j$ are the purely imaginary eigenvalues of the Hamilton map of $Q$ — equality included.

Hörmander's paper itself recorded what its theorem did not cover, in six issues: (1) the equality case of the condition (Theorem 4.3.2 assumes strict inequality); (2) situations where $\Sigma$ is singular or the Hessian type changes along it (a transition, for instance between his types Ic and Ia); (3) the effectively hyperbolic case, excluded as already treated by other authors; (4) systems (Theorem 3.5.1) only for principal symbols with one-dimensional kernel; (5) the asymptotic-solution construction behind the necessity result (Theorem 1.5.1) only in a rigid geometry, a microlocal version being desirable; (6) the possibility that lower-order terms force a loss of more than $m-2$ derivatives.

Since then, three regimes have crystallized as the surviving open core. First, the equality case: when $\operatorname{Tr}^+F_p=0$ on $\Sigma$ the Ivrii–Petkov condition reduces to the Levi condition $P_{sub}=0$ on $\Sigma$, and Nishitani (2014) proved this necessary and sufficient for local and microlocal $C^\infty$ well-posedness when no null bicharacteristic is tangent to $\Sigma$; the case $\operatorname{Tr}^+F_p>0$ with equality attained is open. Second, spectral transition: the Jordan-block structure of the Hamilton map changes across $\Sigma$; only Gevrey-type sufficiency results in special configurations are known, and Melrose's conjectured necessary condition $\operatorname{dist}_{\mathbb{C}}(P_{sub}(\rho),[-\operatorname{Tr}^+F_p(\rho),\operatorname{Tr}^+F_p(\rho)])\le Ce(\rho)$, where $\pm e(\rho)$, $e(\rho)\ge0$, are the real eigenvalues of $F_p(\rho)$ — which vanish on the non-effectively hyperbolic side of $\Sigma$ and, on the effectively hyperbolic side, are comparable to the distance to the transition locus — and $C$ is uniform on the conic double set, remains unproved. Third, tangent bicharacteristics: null bicharacteristics (integral curves of $H_p$ along which the symbol vanishes simply, away from $\Sigma$) that land tangentially on $\Sigma$; Nishitani (2015) showed these exist generically for non-effectively hyperbolic operators, they obstruct $C^\infty$ well-posedness even under the Levi condition, and only model-class results are known. Gevrey classes $G^s$ — functions whose derivatives satisfy $|\partial^\alpha u|\le C^{1+|\alpha|}(\alpha!)^s$, with $s=1$ real analyticity and larger $s$ weaker regularity — provide the natural scale between analytic and $C^\infty$ in which these obstructions are quantified.

Problem Statement

Complete the well-posedness theory of the Cauchy problem for hyperbolic operators with double characteristics in the regimes excluded by Hörmander's 1977 theorem. The common objective is a necessary-and-sufficient characterization of local and microlocal well-posedness (in $C^\infty$ or, where that is impossible, in the sharp Gevrey class) that matches the invariant, non-strict Ivrii–Petkov condition on the subprincipal symbol and allows the spectral type of the Hamilton map to vary along the double set. The source question decomposes into the following separate but linked open questions, all bearing on that objective.

1. Equality case. Assume $\Sigma$ is a $C^\infty$ manifold with the constant rank and constant type of Hörmander's theorem, the operator is non-effectively hyperbolic with $\operatorname{Tr}^+F_p>0$ on $\Sigma$, and the subprincipal symbol satisfies only the non-strict condition

$-\operatorname{Tr}^+F_p\ \le\ P_{sub}\ \le\ \operatorname{Tr}^+F_p\quad\text{on }\Sigma.$

Is the Cauchy problem then well posed in $C^\infty$, possibly with a fixed finite loss of derivatives? If not, in which Gevrey classes is it well posed, and does the non-strict condition alone suffice?

2. Spectral transition and singular double sets. Determine well-posedness when the symplectic type of the Hessian of the principal symbol, equivalently the Jordan-block structure of the Hamilton map, changes across $\Sigma$, or when $\Sigma$ fails to be a smooth constant-rank manifold. In particular, prove or refute Melrose's (1984) conjectured necessary condition $\operatorname{dist}_{\mathbb{C}}(P_{sub}(\rho),[-\operatorname{Tr}^+F_p(\rho),\operatorname{Tr}^+F_p(\rho)])\le Ce(\rho)$ in transition regimes, where $\pm e(\rho)$ ($e(\rho)\ge0$) are the real eigenvalues of $F_p(\rho)$, the inequality is required for $\rho\in\Sigma$ with a constant $C$ uniform on the conic double set, and give a complete necessary-and-sufficient criterion for well-posedness across such transition loci.

3. Tangent bicharacteristics. In the regime where null bicharacteristics land tangentially on $\Sigma$, determine the sharp Gevrey thresholds for well-posedness as a function of the geometry of the tangency, and decide whether any $C^\infty$ theory survives. This regime meets the equality case at $\operatorname{Tr}^+F_p=0$: there the sufficiency direction of Nishitani's (2014) result — the Levi condition $P_{sub}=0$ on $\Sigma$ implying $C^\infty$ well-posedness — is known to fail: with tangent bicharacteristics, $C^\infty$ well-posedness can fail even when the Levi condition holds.

4. Systems with higher-dimensional kernel. Extend the systems theory (Hörmander's Theorem 3.5.1) to principal symbols with kernel dimension greater than one.

5. Sharpness of the $m-2$ derivative loss. Determine whether lower-order terms can force a loss of more than $m-2$ derivatives under the hypotheses of Hörmander's theorem.

References

1. Lars Hörmander, 'The Cauchy problem for differential equations with double characteristics', Journal d'Analyse Mathématique 32 (1977) 118–196, DOI 10.1007/BF02803578, https://doi.org/10.1007/BF02803578
2. V. Ya. Ivrii and V. M. Petkov, 'Necessary conditions for the Cauchy problem for non-strictly hyperbolic equations to be well-posed', Russian Mathematical Surveys 29(5) (1974) 1–70, DOI 10.1070/RM1974v029n05ABEH001295, https://doi.org/10.1070/RM1974v029n05ABEH001295
3. Richard Melrose, 'The Cauchy problem for effectively hyperbolic operators', Hokkaido Mathematical Journal 12 (1983) 371–391, DOI 10.14492/hokmj/1525852964, https://doi.org/10.14492/hokmj/1525852964
4. Enrico Bernardi and Tatsuo Nishitani, 'On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 5 well-posedness', Journal d'Analyse Mathématique 105 (2008) 197–240, DOI 10.1007/s11854-008-0035-3, https://doi.org/10.1007/s11854-008-0035-3
5. Enrico Bernardi and Tatsuo Nishitani, 'On the Cauchy problem for noneffectively hyperbolic operators: The Gevrey 4 well-posedness', Kyoto Journal of Mathematics 51 (2011) 767–810, DOI 10.1215/21562261-1424857, https://doi.org/10.1215/21562261-1424857
6. Enrico Bernardi and Tatsuo Nishitani, 'On the Cauchy problem for non-effectively hyperbolic operators: the Gevrey 3 well-posedness', Journal of Hyperbolic Differential Equations 8 (2011) 615–650, DOI 10.1142/S0219891611002512, https://doi.org/10.1142/S0219891611002512
7. Cesare Parenti and Alberto Parmeggiani, 'On the Cauchy Problem for Hyperbolic Operators with Double Characteristics', Communications in Partial Differential Equations 34 (2009) 837–888, DOI 10.1080/03605300902892360, https://doi.org/10.1080/03605300902892360
8. Enrico Bernardi, Cesare Parenti and Alberto Parmeggiani, 'The Cauchy Problem for Hyperbolic Operators with Double Characteristics in Presence of Transition', Communications in Partial Differential Equations 37 (2012) 1315–1356, DOI 10.1080/03605302.2012.668258, https://doi.org/10.1080/03605302.2012.668258
9. Tatsuo Nishitani, 'Local and microlocal Cauchy problem for non-effectively hyperbolic operators', Journal of Hyperbolic Differential Equations 11 (2014) 185–213, DOI 10.1142/S0219891614500052, https://doi.org/10.1142/S0219891614500052
10. Tatsuo Nishitani, 'A simple proof of the existence of tangent bicharacteristics for noneffectively hyperbolic operators', Kyoto Journal of Mathematics 55 (2015) 281–297, DOI 10.1215/21562261-2871758, https://doi.org/10.1215/21562261-2871758
11. Bernard Lascar and Richard Lascar, 'Une condition suffisante pour que le problème de Cauchy soit bien posé', Israel Journal of Mathematics 211 (2016) 1–12, DOI 10.1007/s11856-015-1269-2, https://doi.org/10.1007/s11856-015-1269-2
12. Tatsuo Nishitani, 'Cauchy Problem for Differential Operators with Double Characteristics', Lecture Notes in Mathematics, Springer (2017), DOI 10.1007/978-3-319-67612-8, https://doi.org/10.1007/978-3-319-67612-8
13. Tatsuo Nishitani, 'On the Cauchy Problem for Differential Operators with Double Characteristics, A Transition from Non-effective to Effective Characteristics', Publications of the Research Institute for Mathematical Sciences 54 (2018) 317–349, DOI 10.4171/prims/54-2-3, https://doi.org/10.4171/prims/54-2-3
14. Tatsuo Nishitani and Vesselin Petkov, 'Cauchy problem for effectively hyperbolic operators with triple characteristics', Journal de Mathématiques Pures et Appliquées 123 (2019) 201–228, DOI 10.1016/j.matpur.2018.06.006, https://doi.org/10.1016/j.matpur.2018.06.006
15. Tatsuo Nishitani, 'A note on the Cauchy problem for $-D_0^2+2x_1D_0D_2+D_1^2+x_1^3D_2^2+\sum_{j=0}^2b_jD_j$', preprint (2022), arXiv:2208.07534, https://arxiv.org/abs/2208.07534
16. Enrico Bernardi and Tatsuo Nishitani, 'Geometric results for hyperbolic operators with spectral transition of the Hamilton map', preprint (2025), arXiv:2505.21078, https://arxiv.org/abs/2505.21078
17. Enrico Bernardi and Tatsuo Nishitani, 'Notes on tangent bicharacteristics and ill-posedness of the Cauchy problem', preprint (2025), arXiv:2506.08481, https://arxiv.org/abs/2506.08481
18. Tatsuo Nishitani, 'A completion of our earlier work on the Cauchy problem for non-effectively hyperbolic operators', preprint (2026), arXiv:2602.23591, https://arxiv.org/abs/2602.23591
19. Enrico Bernardi, 'Lack of Gevrey solvability for a model operator', preprint (2026), arXiv:2605.27054, https://arxiv.org/abs/2605.27054
20. Richard Melrose, 'The Cauchy problem and propagation of singularities', in Seminar on Nonlinear Partial Differential Equations (S. S. Chern, ed.), Mathematical Sciences Research Institute Publications 2, Springer (1984) 185–201.