Hörmander's open issues on non-uniqueness for the Cauchy problem: strictly convex support geometry, the Hölder regime $0<\lambda\le 1$, and the elliptic case

Hörmander's 1975 chapter 'Non-uniqueness for the Cauchy Problem' (Lecture Notes in Mathematics 459) analyzed the scope of the classical counterexample constructions by determining when the perturbed constant-coefficient equation $P(D)u+a(x)\,Q(D)u=0$ admits nontrivial smooth solutions supported in a half-space $H_N$, with the perturbation $a$ vanishing on the unperturbed side. It characterized non-uniqueness in the analytic and $C^\infty$ regimes and proved non-uniqueness for finitely differentiable coefficients when the minimal exponent $\lambda$ in an anisotropic estimate exceeds $1$, but explicitly left three issues open: (1) whether the constructions can be modified so that the solution is supported in a strictly convex set, compact sets being 'particularly important', which would allow the direct comparison with uniqueness theorems that assume vanishing outside strictly convex sets; (2) the regime $0<\lambda\le 1$ of the anisotropic estimate governing finite-regularity perturbations, where the source states the answer is not obvious; and (3) the elliptic case, where the smooth perturbation criterion is vacuous below the analytic threshold and only Pliś-type counterexamples are known. All three issues were left open in the source and remain open.

Background

The Cauchy problem for a linear partial differential equation asks whether a solution is determined by the data — the values of the solution and finitely many of its derivatives — prescribed on an initial hypersurface. Its uniqueness aspect is equivalent to unique continuation: if a smooth solution vanishes on one side of a hypersurface, and hence with all derivatives on the hypersurface itself, must it vanish identically? For operators with analytic coefficients this is settled by Holmgren's theorem, which guarantees uniqueness whenever the initial surface is non-characteristic, meaning that it is not tangent to the characteristic set of the operator's principal symbol (the highest-order part of the operator, read as a polynomial in the frequency variable). For non-analytic coefficients uniqueness can fail: De Giorgi (1955) gave an example for a parabolic equation, Cohen (1960) constructed a smooth planar vector-field equation with a nontrivial solution supported in a half-space, and Pliś constructed elliptic equations with smooth complex-valued coefficients admitting such solutions (second-order case 1963).

Hörmander's 1975 chapter 'Non-uniqueness for the Cauchy Problem' set out, in his words, to 'narrow the gap or at least make it well defined' between these counterexamples and the available uniqueness theorems, by making a systematic analysis of the scope of the constructions used in the counterexamples. The framework isolates the perturbative mechanism common to the known examples: $P(D)$ and $Q(D)$ are differential operators with constant coefficients on $\mathbb{R}^n$ (with $D=-i\,\partial/\partial x$), $H_N=\{x:\langle x,N\rangle\ge 0\}$ is a half-space with normal $N$, and one asks when

$$P(D)u+a(x)\,Q(D)u=0$$

has a solution $u\in C^\infty(\mathbb{R}^n)$ with $\operatorname{supp}u\subset H_N$, the coefficient $a$ being required to vanish where $\langle x,N\rangle\le 0$ so that the operator $P(D)$ is unperturbed on that side. An elliptic operator is one whose principal symbol $P_m(\xi)$ vanishes only at $\xi=0$, and $\operatorname{ord}P$ denotes the order of $P$.

The main results locate the uniqueness/non-uniqueness boundary according to the regularity of $a$. If $a$ is real analytic and the boundary of the half-space is non-characteristic, non-uniqueness holds exactly when $\operatorname{ord}Q>\operatorname{ord}P$. If $a$ is merely $C^\infty$, the criterion involves an anisotropic majorant of the symbol: writing

$$\widetilde{P}_N(\xi)=\Big(\sum_{j\ge 0}|\langle D,N\rangle^jP(\xi)|^2\Big)^{1/2}$$

for the $\ell^2$-norm of the directional derivatives of the symbol $P$ in the normal direction (and $\widetilde{Q}_N$ analogously), non-uniqueness with $C^\infty$ coefficients holds precisely when the quotient $\widetilde{Q}_N(\xi)/\widetilde{P}_N(\xi)$ is unbounded. A companion characterization in terms of limits along complex frequencies $\zeta$ in the region $\sum_N=\{\xi+isN:s\le 0\}$ covers the intermediate structure. For finitely differentiable $a$ the relevant hypothesis is an estimate

$$\widetilde{Q}_N(\zeta,T)\le C\,\widetilde{P}_N(\zeta,T)\Big(1+\frac{|\operatorname{Im}\zeta|}{T}\Big)^{\lambda},$$

with a $T$-anisotropic version $\widetilde{P}_N(\zeta,T)$ of the majorant, required for all $T\ge\sqrt{1+|\operatorname{Im}\zeta|}$; when the minimal such exponent satisfies $\lambda>1$, the chapter constructs a coefficient $a$ belonging to every Hölder class $H^\mu$ with $\mu<\lambda$ (here $H^\mu$ is the Hölder space of functions whose derivatives of order $\lfloor\mu\rfloor$ are $(\mu-\lfloor\mu\rfloor)$-Hölder continuous) and satisfying the boundary decay $|D^\alpha a(x)|\le C_{\varepsilon,\alpha}\langle x,N\rangle^{\lambda-|\alpha|-\varepsilon}$, for which the equation has a nontrivial smooth solution with $\operatorname{supp}u=H_N$. The exponent $\lambda=1$ sits at the Lipschitz threshold of regularity.

The chapter itself delimits what the constructions do not reach, in three explicit remarks. Geometry: 'Most uniqueness theorems known for equations of the form (1.1) require that u vanishes outside a set with a strictly convex boundary. A direct comparison with the counterexamples proved here is therefore not possible. For this and other reasons it would be interesting to modify the constructions with $H_N$ replaced by a strictly convex set, compact sets being particularly important. However, we shall not consider this problem at all here.' Regularity: the Hölder-regime theorem covers $\lambda>1$, while for $0<\lambda\le 1$ the chapter states the answer is not obvious — some examples still fit the non-uniqueness mechanism, while in other cases uniqueness may persist. Ellipticity: when $P(D)$ is elliptic, the $C^\infty$ majorant criterion is vacuous except for the higher-order perturbations $\operatorname{ord}Q>\operatorname{ord}P$ already covered by the analytic case, and the only known smooth elliptic counterexamples, of Pliś type, are obtained by ideas of a different nature from the chapter's exponential constructions.

The direct descendants of the chapter are the 1980s–1990s counterexample and uniqueness literature for variable-coefficient operators (Alinhac, Lerner, Colombini–Del Santo, Alinhac–Baouendi, and others) and, on the uniqueness side, the partial-analyticity theorems of the late 1990s (Hörmander 1997) and their Gevrey refinements; a Gevrey class $G^s$, $s\ge 1$, consists of the $C^\infty$ functions whose derivatives of order $k$ are bounded by $C^{k+1}(k!)^s$, the case $s=1$ being the analytic scale. On the counterexample side, Gevrey-class counterexamples for wave equations (Alinhac–Baouendi 1995; Hörmander 2000; Guisset–Shao 2024) still produce solutions supported in a half-space. A parallel program for elliptic equations in divergence form — equations $-\operatorname{div}(A(x)\nabla u)=0$ with matrix-valued $A$, as opposed to the general non-divergence form of Pliś's example — delimited the coefficient-regularity threshold for unique continuation (Miller 1974; Mandache 1998; Jeznach 2025). Weak unique continuation means that a solution vanishing on an open set vanishes identically; strong unique continuation means that vanishing to infinite order at a single point already forces vanishing. None of these lines settles the three flagged issues within the chapter's framework.

Problem Statement

This problem records the three issues that Hörmander (1975) explicitly left open in the analysis of non-uniqueness for the perturbed constant-coefficient Cauchy problem

$$P(D)u+a(x)\,Q(D)u=0,\qquad \operatorname{supp}u\subset H_N,$$

where $P(D)$ and $Q(D)$ are constant-coefficient differential operators on $\mathbb{R}^n$, $H_N=\{x:\langle x,N\rangle\ge 0\}$ is a half-space, and the perturbation $a$ vanishes where $\langle x,N\rangle\le 0$ so that $P(D)$ is unperturbed there. The common objective, in the source's own formulation, is to 'narrow the gap or at least make it well defined' between the counterexample constructions and the uniqueness theorems — that is, to complete the delineation of when the equation admits nontrivial smooth solutions supported in the prescribed set, as a function of the support geometry, the regularity of $a$, and the structure of the operators. The three unresolved axes are:

(1) Support geometry. Determine whether the non-uniqueness constructions can be modified with the half-space $H_N$ replaced by a strictly convex set, compact sets being particularly important: either exhibit operators $P,Q$ and a coefficient $a$ (in one of the regularity classes covered by the half-space theory) for which the equation admits a nontrivial $C^\infty$ solution supported in a strictly convex set (in particular a compact one), with $a$ vanishing in the exterior in the analogous sense; or prove that support in such a set forces uniqueness under hypotheses comparable to those of the half-space theory. The source flags this as exactly the obstruction to a direct comparison with the uniqueness theorems, which require vanishing outside sets with strictly convex boundary.

(2) The regime $0<\lambda\le 1$ of the anisotropic estimate. The chapter proves non-uniqueness for finitely differentiable $a$ when the minimal exponent $\lambda$ for which the anisotropic estimate of the Hölder-regime theorem holds exceeds $1$, the construction then yielding $a\in\bigcap_{\mu<\lambda}H^\mu$; for $0<\lambda\le 1$ the source states the answer is not obvious. Characterize that regime: determine in which cases non-uniqueness persists (by the chapter's mechanism or another) and in which cases uniqueness holds.

(3) The elliptic case. For elliptic $P(D)$ (principal symbol vanishing only at the origin), the $C^\infty$ criterion of the majorant theorem is vacuous except for the higher-order perturbations $\operatorname{ord}Q>\operatorname{ord}P$ already covered by the analytic case, and the only known smooth elliptic counterexamples are of Pliś type, obtained by ideas of a different nature. Determine the uniqueness/non-uniqueness boundary for perturbations $a\,Q(D)$ of elliptic constant-coefficient operators within this framework.

Under hypotheses comparable to the source's, an in-which-cases boundary statement is two-sided and therefore requires both a counterexample construction and a uniqueness theorem; the overall objective, as in the source, is the sharp delineation of the scope of the counterexample constructions along the three axes.

References

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