Open core of Hörmander's hyperbolic systems with double characteristics: complex stability without the real-determinant hypothesis, classification in general size and dimension, extremal counts, codimension beyond the interpolation regime, and well-posedness for scalar-type double characteristics

In 1993 Lars Hörmander determined, in the lowest dimensions, the local structure of constant-coefficient hyperbolic systems whose principal symbol has non-degenerate double characteristics: points of the characteristic surface where the determinant vanishes quadratically with Hessian of maximal rank. For real $3\times 3$ first-order systems he derived the normal form descending from John's modified elasticity example (four non-degenerate double characteristics, hyperbolicity stratum of codimension 4); for complex hermitian $3\times 3$ systems in five variables, the normal form with six double characteristics (complex codimension 28); for second-order $3\times 3$ systems, explicit constructions with twelve non-degenerate double characteristics. Stability of such double characteristics under hyperbolic perturbation is forced, in the real $2\times 2$ model, by the Moebius-band topology of the kernel line bundle, and in the complex $2\times 2$ model by a non-trivial complex line bundle over the two-sphere -- but the complex stability theorem requires the perturbed determinant to be real-valued. The codimension formula $\operatorname{codim}=\nu$ is proved only under a surjectivity condition on the evaluation of quadratic forms that is impossible for more than ten double characteristics, and the $C^\infty$ well-posedness theorem for the Cauchy problem assumes kernel dimension 2 with Hessian rank at least 3.

Background

A linear system of partial differential equations with constant coefficients, $P(\partial)u=f$, is specified by its symbol: a $k\times k$ matrix $P(\xi)$ of homogeneous polynomials of degree $m$ in the frequency variables $\xi=(\xi_1,\dots,\xi_n)$, obtained by replacing each derivative by $i\xi_j$ (first-order systems: $m=1$, linear entries; second-order systems: $m=2$, quadratic entries). The system is hyperbolic with respect to a real time covector $N$ (say the $\tau$-axis, $\tau=\xi_0$) if the characteristic polynomial $\det P(\tau,\xi)$ has, for every real spatial frequency $\xi$, only real roots in $\tau$ -- that is, $\det P$ is a hyperbolic polynomial in $\tau$. The condition constrains only the roots: no reality is imposed on the coefficients of $\det P$ or of $P$, so the definition applies verbatim to symbols with complex entries, for which $\det P(\tau,\xi)$ is then a complex-valued polynomial in $\tau$ all of whose roots are required to be real for every real $\xi$. Hyperbolicity is the algebraic condition under which the Cauchy problem (solve the equation for $u$ given initial data on a time slice) is expected to be solvable. It is strict when the roots in $\tau$ are distinct for every $\xi\ne 0$; strict hyperbolicity implies well-posedness (existence, uniqueness, continuous dependence on the data) via energy estimates constructed from a symmetrizer -- a smooth, positive definite Hermitian matrix function $S(\xi)$ for which $S(\xi)P(\xi)$ is Hermitian. When roots have multiplicity (weak hyperbolicity), well-posedness can fail or require extra structure, and this is the classical borderline the present problem addresses.

A characteristic direction at which a root has multiplicity exactly two is a double characteristic: a point of the characteristic surface $\{\det P=0\}$ that is critical for $\det P$, so the determinant vanishes to second order. There the Hessian (matrix of second derivatives) of $\det P$ controls the local geometry; hyperbolicity constrains its signature, and the double characteristic is called non-degenerate when the Hessian has maximal rank -- rank 3 at a non-zero double characteristic of the four-variable systems studied here, meaning $\det P$ is locally a quadratic cone in the three transverse variables. (The Hessian of a homogeneous polynomial at a non-zero critical point always annihilates the radial direction, so its rank there is at most $n-1$; at the origin the radial vector is zero, no such constraint holds, and the Hessian may have the full rank $n$, as for the $2\times 2$ models below, whose double characteristic is at the origin.) The integer $\dim\ker P(\xi^0)$, the dimension of the kernel of the matrix symbol at the double point, separates system-type degeneracies ($\dim\ker P=2$: two characteristic sheets crossing transversally in a cone, the pattern of conical refraction -- the splitting of a light ray into a hollow cone inside a biaxial crystal -- in crystal optics) from scalar-type ones ($\dim\ker P=1$: the matrix itself degenerating in a Jordan-type fashion, possible only for order $m\ge 2$ or non-symmetric symbols). Fritz John showed in 1978, via a modified system of three-dimensional elasticity, that hyperbolic $3\times 3$ first-order systems can carry four non-degenerate double characteristics that cannot be removed: near his example, the set of systems hyperbolic with respect to the given covector is a stratum of codimension 4 in coefficient space, and every system in that stratum retains the double characteristics.

Hoermander's 1993 paper made the phenomenon systematic. For real $2\times 2$ models (symbol $Q$ with $\det Q=\eta_1^2-\eta_2^2-\eta_3^2$, whose doubly characteristic set is the origin -- a codimension-3 point, where $Q=0$ and the kernel is two-dimensional -- and whose characteristic set $\{\det Q=0,\ \eta\ne 0\}$ is a codimension-1 cone) he identified the topological mechanism: the kernels of the symbol along the characteristic cone assemble into a real line bundle -- a family of one-dimensional vector spaces, one over each point of the base space -- which is a Moebius strip, and its non-triviality (the strip does not untwist into a cylinder) forces persistence of the double characteristics; the complex $2\times 2$ model exhibits instead a non-trivial complex line bundle over the two-sphere, glued from two trivial hemispheric halves by a clutching function of winding number $-1$ (the winding number of this gluing map is the integer invariant classifying complex line bundles over the sphere). From the $2\times 2$ models he derived normal forms: the John-type form for $3\times 3$ real first-order systems (four non-degenerate double characteristics, codimension 4), and, for hermitian complex $3\times 3$ systems in five variables, a form with six double characteristics subject to a triangle condition on its parameters, the stratum having complex codimension 28. For second-order $3\times 3$ systems he constructed, via symbols $A(\xi)=C|\xi|^2 I+\sum_j q_j(\xi)(b^j\otimes b^j)$ with suitable quadratic forms $q_j$ and vectors $b^j$ ($\otimes$ the outer product), systems with twelve non-degenerate double characteristics. He proved the codimension of the stratum of systems with $\nu$ non-degenerate double characteristics equals $\nu$ whenever the evaluation map $q\mapsto(q(\xi^1),\dots,q(\xi^\nu))$ on the space of quadratic forms in four variables (a ten-dimensional space) is surjective -- which can hold only for $\nu\le 10$. His complex stability theorem requires, however, that the perturbed determinant also be real-valued. Finally, for the Cauchy problem he proved: if the doubly characteristic set is a manifold of codimension 3 on which $\dim\ker P=2$ and the Hessian of $\det P$ has rank at least 3, then $P$ satisfies the resolvent estimate $|\tau|(|\xi|+|\tau|)^{m-1}|P(\xi+i\tau N)^{-1}|\le C$ in a complex half-space, and the Cauchy problem is $C^\infty$ well-posed for arbitrary lower-order terms.

The surrounding literature approaches this border from other sides. The symmetrizer school (Nishitani; Vaillant; Metivier) studies which hyperbolic systems admit smooth symmetrizers and hence $L^2$ well-posedness; Nishitani's 2014 monograph chapter defines non-degeneracy as maximal Hessian rank of the determinant and proves symmetrizability and well-posedness results for systems with analytic coefficients in that setting. Braam and Duistermaat (1993) classify real symmetric systems with multiplicity by group-theoretic normal forms, with variable-coefficient and physics applications (Maxwell equations, elastodynamics). Garetto, Jaeh and Ruzhansky (2018) treat variable-coefficient systems with non-diagonalisable principal symbol. The scalar-operator theory of effectively and non-effectively hyperbolic operators with double characteristics -- where the spectrum of the Hamilton map (the linearization at the double point of the Hamilton flow that propagates singularities along the characteristic surface) governs well-posedness -- was developed in parallel, culminating in Nishitani's 2017 monograph. The geometric line of Arnold (1988) and Khesin (1990) studies singularities of light hypersurfaces and of hyperbolicity sets, continued by Valero with Morse-theoretic methods on Fresnel surfaces (the wave surfaces of crystal optics). Real algebraic geometry contributes the theory of hyperbolic polynomials and their perturbations (linear pencils of hyperbolic polynomials in the work of Kurdyka and Paunescu).

Problem Statement

The objective is to complete the local structure theory, begun by Hoermander (1993), of constant-coefficient hyperbolic systems with isolated non-degenerate double characteristics, and to settle the well-posedness of the Cauchy problem at the border of the known theory. Throughout, $P$ denotes a $k\times k$ matrix of homogeneous polynomials of degree $m$ in $n$ real variables whose determinant is hyperbolic with respect to a real covector $N$; a double characteristic is a characteristic direction $\xi^0$ with $\det P(\xi^0)=0$ and $\operatorname{grad}(\det P)(\xi^0)=0$; it is non-degenerate when the Hessian of $\det P$ at $\xi^0$ has maximal rank. Five linked questions complete the theory at the full generality at which the results of Hoermander (1993) leave it open (no restriction on $k$, $n$, $m$, or symmetry beyond what is written); all five are required to meet the objective, and they are pairwise independent -- no one of them resolves another -- so each must be answered on its own:

(A) Complex stability without the real-determinant hypothesis. Hoermander's complex stability theorem for the $2\times 2$ model covers perturbations $Q_1$ whose determinant is real-valued, showing that a non-degenerate double characteristic near a symbol with $\det Q=\eta_1^2-\eta_2^2-\eta_3^2-\eta_4^2+O(|\eta|^3)$ cannot be removed within the class of hyperbolic systems with real determinant. Determine whether the same stability holds, or fails, for hyperbolic perturbations whose determinant is complex-valued: perturbations $Q_1$ that are hyperbolic in the sense of the Background -- for every real spatial frequency $\xi$, all roots in $\tau$ of the complex polynomial $\det Q_1(\tau,\xi)$ are real -- but for which $\det Q_1$ is not real-valued, so that the real-determinant hypothesis of the theorem is dropped while the notion of hyperbolicity is unchanged; if stability fails, exhibit the removing perturbation.

(B) Classification in general size and dimension. The normal-form classification is complete only for $3\times 3$ real first-order systems in four variables (four non-degenerate double characteristics, codimension 4), hermitian complex $3\times 3$ systems in five variables (six double characteristics, complex codimension 28), and the second-order $3\times 3$ constructions. Extend the classification to general matrix size $k$, number of variables $n$, and order $m$: determine the normal forms of hyperbolic systems with non-removable non-degenerate double characteristics, the strata of the hyperbolicity set through them, and the codimensions of those strata in the interpolation regime $\nu\le n(n+1)/2$ (the dimension of the space of quadratic forms in the $n$ frequency variables, which equals $10$ only at $n=4$); the case $\nu>n(n+1)/2$ beyond that regime is the separate target of (D).

(C) Extremal count. For each $(k,n,m)$, determine the exact maximal number of isolated non-degenerate double characteristic directions that a hyperbolic $k\times k$ system of order $m$ in $n$ variables can possess; deciding only whether a sharp finite upper bound exists does not by itself resolve this question. Known: 4 (real first-order $3\times 3$), 6 (hermitian complex $3\times 3$ in five variables), 12 (second-order $3\times 3$, by construction); no upper bound is known in any case.

(D) Codimension beyond the interpolation regime. The formula $\operatorname{codim}=\nu$ rests on surjectivity of the evaluation map on the space of quadratic forms in the $n$ frequency variables at the $\nu$ double points, a space of dimension $n(n+1)/2$ (ten only at $n=4$), which is dimensionally impossible once $\nu>n(n+1)/2$. Determine the codimension of the hyperbolicity strata for all systems with $\nu>n(n+1)/2$ non-degenerate double characteristics; the stratum of the twelve-double-characteristic second-order constructions (four variables, $\nu=12$), where no method is currently available, is the first case, and resolving it alone does not complete this question.

(E) Well-posedness at the border of the double-characteristic theory. Hoermander's theorem gives $C^\infty$ well-posedness for arbitrary lower-order terms when $\dim\ker P=2$ on the doubly characteristic set and the Hessian rank is at least 3. Decide well-posedness for hyperbolic systems whose double characteristics are scalar-type ($\dim\ker P=1$) or have Hessian rank at most 2 (below the rank-3 hypothesis of the theorem; for the four-variable systems, rank 2 or 1): prove well-posedness under appropriate conditions, or construct ill-posed examples.

Two further directions -- the comparison with the Arnold-Khesin singularity-theoretic description of hyperbolicity sets, and the propagation of singularities and polarization (the fine structure of the wave front set, the microlocal set of cotangent directions in which a solution is singular) -- are not part of the five-question deliverable.

References

1. Lars Hörmander, "Hyperbolic systems with double characteristics", Communications on Pure and Applied Mathematics 46 (1993), no. 2, 261-301. DOI: 10.1002/cpa.3160460207, https://doi.org/10.1002/cpa.3160460207.
2. Fritz John, "Algebraic conditions for hyperbolicity of systems of partial differential equations", Communications on Pure and Applied Mathematics 31 (1978), no. 1, 89-106. DOI: 10.1002/cpa.3160310105, https://doi.org/10.1002/cpa.3160310105.
3. Fritz John, "Addendum to: Algebraic conditions for hyperbolicity of systems of partial differential equations", Communications on Pure and Applied Mathematics 31 (1978), no. 6, 787-793. DOI: 10.1002/cpa.3160310607, https://doi.org/10.1002/cpa.3160310607.
4. V. I. Arnold, "On the interior scattering of waves, defined by hyperbolic variational principles", Journal of Geometry and Physics 5 (1988), no. 3, 305-315. DOI: 10.1016/0393-0440(88)90027-7, https://doi.org/10.1016/0393-0440(88)90027-7.
5. B. A. Khesin, "Singularities of light hypersurfaces and structure of hyperbolicity sets for systems of partial differential equations", in: Theory of Singularities and its Applications, Advances in Soviet Mathematics 1, American Mathematical Society, 1990, pp. 105-118. zbMATH: 0747.35019, https://zbmath.org/?q=an%3A0747.35019. [No DOI.]
6. Enrico Bernardi and Tatsuo Nishitani, "Remarks on symmetrization of 2x2 systems and the characteristic manifolds", Osaka Journal of Mathematics 29 (1992), no. 1, 129-134. zbMATH: 0764.35058, https://zbmath.org/?q=an%3A0764.35058. [No DOI.]
7. Peter J. Braam and J. J. Duistermaat, "Normal forms of real symmetric systems with multiplicity", Indagationes Mathematicae 4 (1993), no. 4, 407-421. DOI: 10.1016/0019-3577(93)90011-M, https://doi.org/10.1016/0019-3577(93)90011-M.
8. A. Shadi Tahvildar-Zadeh, "Relativistic and nonrelativistic elastodynamics with small shear strains", Annales de l'Institut Henri Poincaré, Physique Théorique 69 (1998), no. 3, 275-307. zbMATH: 0930.74008, full text: http://www.numdam.org/item/AIHPA_1998__69_3_275_0/.
9. Nils Dencker, "On the propagation of polarization in conical refraction", Duke Mathematical Journal 57 (1988), no. 1, 85-134. DOI: 10.1215/S0012-7094-88-05704-3, https://doi.org/10.1215/S0012-7094-88-05704-3. [Published version of the Lund report.]
10. Nils Dencker, "The propagation of polarization in double refraction", Journal of Functional Analysis 104 (1992), no. 2, 414-468. DOI: 10.1016/0022-1236(92)90006-5, https://doi.org/10.1016/0022-1236(92)90006-5.
11. Tatsuo Nishitani, "Symmetrization of Hyperbolic Systems with Non-degenerate Characteristics", Journal of Functional Analysis 132 (1995), no. 2, 251-272. DOI: 10.1006/jfan.1995.1106, https://doi.org/10.1006/jfan.1995.1106.
12. Tatsuo Nishitani, "Stability of symmetric systems under hyperbolic perturbations", Hokkaido Mathematical Journal 26 (1997), no. 3, 509-527. DOI: 10.14492/HOKMJ/1351258263, https://doi.org/10.14492/HOKMJ/1351258263.
13. Tatsuo Nishitani, "Systems with Nondegenerate Characteristics", Chapter 4 (pp. 161-229) of: Hyperbolic Systems with Analytic Coefficients: Well-posedness of the Cauchy Problem, Lecture Notes in Mathematics 2097, Springer, 2014. DOI: 10.1007/978-3-319-02273-4_4, https://doi.org/10.1007/978-3-319-02273-4_4.
14. Jean Vaillant, "Complex strongly hyperbolic 4x4 systems, reduced dimension and hermitian systems", Bulletin des Sciences Mathématiques 129 (2005), no. 5, 415-456. DOI: 10.1016/j.bulsci.2005.01.001, https://doi.org/10.1016/j.bulsci.2005.01.001.
15. Tatsuo Nishitani and Jean Vaillant, "Smoothly symmetrizable complex systems and the real reduced dimension", Tsukuba Journal of Mathematics 30 (2006), no. 2, 259-271. DOI: 10.21099/tkbjm/1496165064, https://doi.org/10.21099/tkbjm/1496165064.
16. Guy Métivier, "L^2 well-posed Cauchy problems and symmetrizability of first order systems", Journal de l'École polytechnique — Mathématiques 1 (2014), 39-70. DOI: 10.5802/jep.3, https://doi.org/10.5802/jep.3.
17. Claudia Garetto, Christian Jäh and Michael Ruzhansky, "Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness", Mathematische Annalen 372 (2018), no. 3-4, 1597-1629. DOI: 10.1007/s00208-018-1672-1, https://doi.org/10.1007/s00208-018-1672-1; preprint arXiv:1801.03573, https://arxiv.org/abs/1801.03573.
18. Tatsuo Nishitani, Cauchy Problem for Differential Operators with Double Characteristics, Lecture Notes in Mathematics 2202, Springer, 2017. DOI: 10.1007/978-3-319-67612-8, https://doi.org/10.1007/978-3-319-67612-8.
19. Tatsuo Nishitani, "Transversally strictly hyperbolic systems", Kyoto Journal of Mathematics 60 (2020), no. 4, 1399-1418. DOI: 10.1215/21562261-2019-0066, https://doi.org/10.1215/21562261-2019-0066; preprint arXiv:1704.02558, https://arxiv.org/abs/1704.02558.
20. Carlos Valero, "Maxwell's equations, the Euler index, and Morse theory", Mathematical Notes 100 (2016), no. 3-4, 352-362. DOI: 10.1134/S0001434616090029, https://doi.org/10.1134/S0001434616090029; preprint arXiv:1311.0569, https://arxiv.org/abs/1311.0569.
21. Carlos Valero, "Singularities of the eigenvalue functions for first order symmetric symbols on rank two vector bundles over surfaces", preprint arXiv:1406.3620 (v2, 2019), https://arxiv.org/abs/1406.3620.
22. Krzysztof Kurdyka and Laurentiu Paunescu, "Nuij Type Pencils of Hyperbolic Polynomials", Canadian Mathematical Bulletin 60 (2017), no. 3, 561-570. DOI: 10.4153/CMB-2016-079-1, https://doi.org/10.4153/CMB-2016-079-1; preprint arXiv:1504.03665, https://arxiv.org/abs/1504.03665.