Singular-set structure and classification of minimizers in Newton's problem of minimal resistance

Newton's problem of minimal resistance asks for the convex body of given base and height that experiences the least drag when moving through a rarefied medium; analytically, one minimizes the functional $\int\!\!\int_\Omega(1+|\nabla u|^2)^{-1}dx\,dy$ over concave functions $u:\Omega\to[0,M]$. The nose-stretching method of Plakhov (2021), refined in Plakhov (2022a), shows that every minimizer is the closed convex hull of its singular boundary points — points where the supporting plane is not unique — so the problem reduces entirely to understanding the singular set. Two linked questions remain open. (A) Plakhov's conjecture that the singular set of a minimizer is closed, and therefore nowhere dense: currently not even non-denseness is known, so the regular part of the boundary could in principle degenerate entirely. (B) For the unit circular base, the rigorous classification of minimizers across the height parameter $M$: numerical studies indicate a sequence of bifurcation values $+\infty=M_1>M_2>M_3>\cdots\to0$ with a unique minimizer (up to rotation) whose top level set is a regular $k$-gon for $M_k<M<M_{k-1}$ and two minimizers at $M=M_k$, but no rigorous classification, exact values, or uniqueness theorem exists — and the explicit candidate solutions proposed in the literature have been refuted by the theorem that boundary conical parts destroy optimality. The problem is to prove or disprove (A) and (B).

Background

Newton's problem of minimal resistance, first formulated by Isaac Newton in 1685 (see the historical accounts in Buttazzo–Kawohl 1993 and Buttazzo 2025), asks for the shape of a convex body of given base and height that experiences the least drag when it moves at constant velocity through a rarefied medium of non-interacting particles that rebound elastically, each particle meeting the body at most once. Writing the front surface of the body as the graph of a concave function $u:\Omega\to[0,M]$, where $\Omega\subset\mathbb{R}^2$ is a compact convex base with nonempty interior and $M>0$ is the height, the drag is proportional to the resistance functional

$$R(u)=\int\!\!\int_\Omega\frac{dx\,dy}{1+|\nabla u(x,y)|^2},$$

and the problem is to minimize $R(u)$ over all concave $u$ with $0\le u\le M$ on $\Omega$. The concavity constraint (equivalently, convexity of the body lying below the graph) is what makes the problem nonstandard: it is a pointwise second-order constraint, minimizers are not smooth, and no classical Euler–Lagrange regularity theory applies to them.

Associate with $u$ the convex body $C_u=\{(x,y,z):(x,y)\in\Omega,\ 0\le z\le u(x,y)\}$. A boundary point of $C_u$ is called singular if it admits more than one supporting plane, and regular otherwise. A point of $C_u$ is extreme if it is not a convex combination of two distinct other points of $C_u$. Singular points split into conical points (the tangent cone, i.e. the smallest convex cone with apex at the point that contains the body, is three-dimensional) and ridge points (the tangent cone degenerates into a dihedral angle). A surface patch with identically zero Gaussian curvature is called developable; it is swept out by straight lines. Almost every boundary point of a convex body is regular, and by the Krein–Milman theorem a compact convex body is the convex hull of its extreme points.

What is known about minimizers: existence holds (Buttazzo–Ferone–Kawohl 1995); wherever $\nabla u$ exists, either $\nabla u=0$ or $|\nabla u|\ge1$; the top level set $\{u=M\}$ is neither empty nor a singleton; on any open set where a minimizer is $C^2$ (away from the levels $0$ and $M$) one has $\det D^2u=0$, so smooth patches of the boundary are developable and contain no extreme points (Brock–Ferone–Kawohl 1996) — in particular, minimizers on a disk are not rotationally symmetric, a symmetry-breaking phenomenon; and when the boundary curve of $\Omega$ is regular (for example a circle), every minimizer satisfies $u=0$ on $\partial\Omega$ (Plakhov 2020).

Plakhov (2021) introduced a geometric perturbation technique, the method of nose stretching, and proved that at least one minimizer $u$ satisfies $C_u=\mathrm{Conv}(\overline{\mathrm{Sing}\,C_u})$: the optimal body is the convex hull of the closure of its own singular boundary points, so the boundary is swept by line segments whose endpoints lie in the closure of the singular set. This was strengthened by Plakhov (2022a) to every minimizer: each minimizer is uniquely determined by the singular set of its epigraph, and any open patch of the graph on which a minimizer is $C^1$ contains no extreme points. Consequently all information about a solution is carried by its singular set, and the structure of that set is the central object of the theory. Numerical studies (Lachand-Robert–Oudet 2005, convex-hull method; Wachsmuth 2014; Wachsmuth 2025) indicate that minimizers on the unit disk have dihedral symmetry, polygonal top level sets whose number of sides depends on the height $M$, and lateral boundaries foliated by line segments joining singular curves. However, the explicit candidate bodies proposed in the literature were refuted: Lokutsievskiy–Wachsmuth–Zelikin (2022) proved that any conical part of the boundary destroys optimality, which rules out the conjectured solutions of Lachand-Robert–Peletier (2001) for every $M>0$ and the structural conjecture of Wachsmuth (2014) at $M=1.0$. What remains unknown is precisely what this problem asks about: the topological nature of the singular set of a minimizer, and a rigorous classification of minimizers as the height parameter varies.

Problem Statement

Let $\Omega\subset\mathbb{R}^2$ be a compact convex set with nonempty interior and let $M>0$. Newton's problem of minimal resistance (concave case) is to minimize

$$R(u)=\int\!\!\int_\Omega\frac{dx\,dy}{1+|\nabla u(x,y)|^2}$$

over concave $u:\Omega\to[0,M]$. Write $C_u=\{(x,y,z):(x,y)\in\Omega,\ 0\le z\le u(x,y)\}$ and let $\mathrm{Sing}\,C_u$ denote the set of singular points of $\partial C_u$ (boundary points with more than one supporting plane). Minimizers exist, every minimizer is the closed convex hull of its singular points, and $C^1$ patches of a minimizer's graph contain no extreme points (an extreme point of $C_u$ is a point that is not a convex combination of two distinct other points of $C_u$); what remains open is the characterization of the singular sets themselves, and through them of the minimizers. The target, left open by Plakhov (2021) and restated as open in Plakhov (2022a) and Buttazzo (2025), has two linked components:

(A) Singular-set conjecture. Prove or disprove that the set of singular points of a minimizer is closed, and therefore nowhere dense — in the formulation of Plakhov (2021), $\mathrm{Sing}\,C_u$ is a closed subset of $\partial C_u$ (in the relative topology) for a minimizer $u$; in the formulation of Plakhov (2022a), the set of points of $\Omega$ at which a minimizer fails to be differentiable is a closed nowhere dense subset of $\Omega$. It is currently not even known that the singular set of a minimizer is not dense in the graph.

(B) Classification across the height parameter, for the classical circular base $\Omega=\{(x,y):x^2+y^2\le1\}$. Prove or disprove the taxonomy indicated by numerical computation (Lachand-Robert–Oudet 2005; Wachsmuth 2014; Wachsmuth 2025): there exist bifurcation values $+\infty=M_1>M_2>M_3>\cdots\to0$ such that (i) for $M_k<M<M_{k-1}$, $k\ge2$, the minimizer is unique up to rotation about the $z$-axis and its top level set $\{u=M\}$ is a regular $k$-gon; (ii) for $M=M_k$ there are two distinct minimizers up to rotation, with top level sets a regular $k$-gon and a regular $(k+1)$-gon respectively.

Components (A) and (B) share one scientific objective: the characterization of optimal bodies through their singular sets, which the nose-stretching theory shows to be equivalent to characterizing the minimizers themselves — (A) is the topological regularity of those sets, (B) their classification as the height $M$ varies. The problem is resolved by proving the claims, by refuting them (for instance, by exhibiting a minimizer whose singular set is not closed or is dense in its graph, or a height value at which uniqueness or the polygonal structure of the top level set fails), or by a combination of proof for one component and valid refutation of the other.

References

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