Geometric characterization of the singular supports of solutions of general constant-coefficient partial differential equations and of P-convexity for singular supports

Hörmander's 1972 paper 'On the singularities of solutions of partial differential equations with constant coefficients' introduced the invariant $\sigma_P(V)$ — measuring how much the symbol $P$ grows when its frequency argument is translated inside a linear subspace $V$ at infinity — and used it to characterize exactly which closed convex translation-invariant sets occur as singular supports of solutions of $P(D)u\in C^\infty$, to prove partial propagation-of-singularities results, and to derive Hölder-type estimates. For a general constant-coefficient operator, the open problem is to characterize geometrically (a) the closed sets that arise as singular supports of solutions, (b) the open sets that are P-convex for singular supports — which, together with P-convexity for supports, is by Hörmander (1962) equivalent to surjectivity of $P(D)$ on distributions — and (c) the propagation law that replaces bicharacteristic propagation outside real principal type. The convex translation-invariant case and the special classes are settled (semi-elliptic/subspace-elliptic and the plane on both sides, wave-type principal part on the supports side); Kalmes (2019) records an explicit open $\sigma_P$ question and Ciás–Kalmes (2026) state that even the supports-side analogue is 'far from being solved' in arbitrary dimensions. The problem is open.

Background

A partial differential operator with constant coefficients has the form $P(D)$, where $P$ is a polynomial in $n$ variables with complex coefficients and $D=(D_1,\dots,D_n)$ with $D_j=-i\,\partial/\partial x_j$ is the Fourier-normalized gradient. Solutions are sought in the sense of distributions (continuous linear functionals on the compactly supported smooth functions), and for a distribution $u$ the singular support, written $\operatorname{sing\,supp}u$, is the smallest closed set outside which $u$ is a $C^\infty$ function. A theorem of Malgrange (1956) gives a fundamental solution for every $P\neq 0$, so $P(D)$ is surjective on the space $\mathcal{D}'(\mathbb{R}^n)$ of all distributions; the classical problem is what happens on a proper open domain $\Omega\subseteq\mathbb{R}^n$.

Hörmander (1962) reduced the domain problem to two geometric properties of $\Omega$. An open set $\Omega$ is called P-convex for supports if for every compact $K\subset\Omega$ there is a compact $K'\subset\Omega$ such that $\operatorname{supp}v\subset K'$ whenever $v$ is a compactly supported distribution in $\Omega$ with $\operatorname{supp}P(-D)v\subset K$; P-convexity for singular supports is the same condition with singular supports in place of supports. Hörmander proved that $P(D)$ maps $\mathcal{D}'(\Omega)$ onto $\mathcal{D}'(\Omega)$ if and only if $\Omega$ is P-convex for supports and for singular supports. Both properties are automatic for elliptic $P$ (the top-order part of the symbol vanishes only at the origin) and for convex $\Omega$ (see e.g. Debrouwere–Kalmes 2025), but they are subtle in general. The motivating example is due to John (1960): a solution of the wave equation $u_{tt}-u_{xx}-u_{yy}=0$ in $\mathbb{R}^3$ that is real-analytic in an open cylinder, exactly $C^m$ on the cylinder boundary and $C^{m+1}$ outside — singularities sitting on a curved surface rather than propagating along characteristic planes — with John's own remark that apparently not all types of singularities propagate along characteristic surfaces, as recalled in Hörmander's (1985) commentary on John's work.

Hörmander (1972) built the theory that explains such phenomena; the author's seminar exposition of the same year gives an overview, and his monograph The Analysis of Linear Partial Differential Operators II (Hörmander 2005, §§10.6–10.8 and 11.3) the standard account. The key invariant attaches to each linear subspace $V\subseteq\mathbb{R}^n$ the number

$$\sigma_P(V)=\inf_{t\ge 1}\ \liminf_{|\xi|\to\infty}\frac{\widetilde P_V(\xi,t)}{\widetilde P(\xi,t)},\qquad \widetilde P_V(\xi,t)=\sup_{\theta\in V,\ |\theta|\le t}|P(\xi+\theta)|,\qquad \widetilde P(\xi,t)=\widetilde P_{\mathbb{R}^n}(\xi,t),$$

which measures how much $P$ grows when its frequency argument is translated within $V$ (a bounded-frequency variant $\sigma_P^0$ replaces the limit at infinity by an infimum over $\xi$). Its main results: (i) a closed convex set $\Gamma$ — with $V$ the largest linear subspace such that $\Gamma+V=\Gamma$ — occurs as $\operatorname{sing\,supp}u$ for some distribution $u$ with $P(D)u\in C^\infty$, with any prescribed finite smoothness off $\Gamma$, if and only if $\sigma_P(V^\perp)=0$, $V^\perp$ the orthogonal complement of $V$; (ii) uniqueness of continuation in the spirit of Holmgren's classical uniqueness theorem for equations with analytic coefficients holds in the form: if $W$ is spanned by the gradients of local $C^1$ defining functions of a surface and $\sigma_P(W)\neq 0$, then a solution of $P(D)u\in C^\infty$ that is $C^\infty$ on one side of the surface is $C^\infty$ near it; and partial propagation statements hold under uniform hypotheses on the localizations of $P$ (polynomials obtained from $P$ by translating and dilating the frequency variable); (iii) positivity of $\sigma_P^0$ is equivalent to Hölder-type (geometric-mean) estimates for solutions. Hörmander emphasized that the minimal invariant subspaces $V$ play to a large extent the role that bicharacteristics — the Hamilton-flow curves of the principal symbol — play for operators of real principal type (operators whose principal symbol has non-vanishing gradient on its zero set), but that examples show $\operatorname{sing\,supp}u$ with $P(D)u\in C^\infty$ is not always a union of such subspaces.

For operators of real principal type, the microlocal theory of the wave front set (the refinement of the singular support that also records the frequency directions causing the singularity) yields a complete propagation law along bicharacteristics, but for constant-coefficient operators of non-principal type the 1972 theory is still the frontier. What is missing is the geometric side: characterizations of P-convexity in terms of the shape of $\Omega$ alone. An open set is called strongly P-convex if it is P-convex for both supports and singular supports. Known cases: for elliptic $P$ every open set is strongly P-convex; for semi-elliptic/subspace-elliptic operators (operators that act along a subspace $W$ — the symbol is constant in the $W^\perp$ directions — and are elliptic on $W$) P-convexity is characterized by minimum principles for the boundary-distance function $d_\Omega(x)=\operatorname{dist}(x,\Omega^c)$, i.e. for every compact $K\subset\Omega\cap L$, $L$ such an affine subspace, the minimum of $d_\Omega$ over $K$ is attained on the relative boundary of $K$ in $L$ (Kalmes 2019); likewise for P-convexity for supports when $P$ is a second-order operator whose principal part is the wave operator (Hörmander 2005, Thm 10.8.6, essentially due to Persson); and in the plane $\mathbb{R}^2$, where P-convexity for supports is characterized (Hörmander 2005, Thm 10.8.3) and Kalmes (2011) proved that every P-convex-for-supports open subset of $\mathbb{R}^2$ is already strongly P-convex, settling the singular-supports side as well. The general case is open and is the subject of this problem.

Problem Statement

Let $P$ be a non-constant polynomial in $n$ variables with complex coefficients, $P(D)$ the associated constant-coefficient operator on distributions in $\mathbb{R}^n$, and for a linear subspace $V\subseteq\mathbb{R}^n$ let $V^\perp$ denote the orthogonal complement and $\sigma_P(V)$ the invariant of Hörmander (1972) defined in the Background above. Hörmander (1972) settled the convex translation-invariant case: for a closed convex set $\Gamma$, with $V$ the largest linear subspace such that $\Gamma+V=\Gamma$, one has $\Gamma=\operatorname{sing\,supp}u$ for a solution of $P(D)u\in C^\infty$ (with prescribed finite smoothness off $\Gamma$) if and only if $\sigma_P(V^\perp)=0$; the paper also proves partial propagation results — Holmgren-type theorems when $\sigma_P(W)\neq 0$, and propagation for compact families of subspaces under uniform hypotheses on the localizations of $P$ — and its examples show that singular supports need not be unions of linear subspaces. The open problem is to complete the theory to general $P$:

(a) Set level. Characterize, for arbitrary $P$, the closed subsets $F\subseteq\mathbb{R}^n$ that occur as $\operatorname{sing\,supp}u$ for some distribution $u$ with $P(D)u\in C^\infty$ — beyond the convex translation-invariant sets settled in 1972.

(b) Domain level. Characterize geometrically, for arbitrary $P$, the open sets $\Omega\subseteq\mathbb{R}^n$ that are P-convex for singular supports (meaning: for every compact $K\subset\Omega$ there is a compact $K'\subset\Omega$ such that $\operatorname{sing\,supp}v\subset K'$ whenever $v$ is a compactly supported distribution in $\Omega$ with $\operatorname{sing\,supp}P(-D)v\subset K$) — by conditions on the shape of $\Omega$ (in the spirit of minimum principles for the boundary-distance function in operator-dependent affine subspaces), rather than by the algebraic $\sigma_P$/localization conditions. Here $\sigma_P$ (or the characteristic variety) may be used to identify the relevant affine subspaces; but the resolving condition must be a shape property of $\Omega$ — a minimum-principle/convexity condition on $\Omega$ — and not merely the statement $\sigma_P(\cdot)=0$ or a list of localizations of $P$. By Hörmander (1962), together with P-convexity for supports this is equivalent to surjectivity of $P(D)$ on $\mathcal{D}'(\Omega)$, so the answer would complete the geometric solution of the classical Malgrange–Hörmander surjectivity problem on domains.

(c) Propagation law. Give, for arbitrary $P$, a necessary-and-sufficient propagation theorem for solutions of $P(D)u\in C^\infty$: a law that specialises to the bicharacteristic propagation law when $P$ is of real principal type, and from which the set-level and domain-level characterisations (a) and (b) follow. A one-directional propagation constraint — of the kind already available from the partial statements of Hörmander (1972) — is therefore not by itself a resolution of (c).

Deliverables (a)–(c) form one scientific objective: a shape-level description of the singular-support geometry of a general constant-coefficient equation. (c) is the mechanism from which (a) and (b) are expected to follow — the 1972 results derive both set-level and domain-level statements from propagation — and (a) and (b) are the global and local readings of the same classification. A complete resolution requires (a), (b) and (c) to be established jointly: establishing (c) alone does not resolve (a)/(b) unless the set- and domain-level statements are actually derived from it, and conversely (a) and (b) do not substitute for (c). The adjacent refinement issues — effective computation of $\sigma_P$ and $\sigma_P^0$ for general $P$, optimal Hölder exponents and constants in the $\sigma_P^0$ estimates, and extension of the $\sigma$-criteria beyond constant strength to variable-coefficient or pseudodifferential operators — are not part of this problem: the first is subsumed by any resolution of (a)/(b), the second is a quantitative refinement of the regularity estimates, and the third is partially addressed for constant-strength operators and largely superseded outside the constant-coefficient setting by microlocal wave-front-set theory.

References

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