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Alexander Plakhov

Publications and source records attributed to Alexander Plakhov.

At least 19 recordsLinked to original sources
arXiv2026

Isoperimetric problem for quasi-Finsler metrics and shapes of least aerodynamic resistance

We consider the following model in the framework of Newtonian aerodynamics: a 2D convex body moves forward and slowly oscillates in a rarefied medium on the plane. The law of oscillations is given. The problem is to find a body of fixed area that has the smallest resistance. We solve this problem by reducing it to an isoperimetric problem for quasi-Finsler metrics. Further, we introduce two criteria of smallness of oscillations. We show that the optimal body has singularity at the front (back) part of its boundary iff the former (latter) criterion is satisfied. The rest of the boundary is smooth. Finally, we find exact optimal shapes in the case of uniform oscillations and construct several shapes explicitly.

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arXiv2025

On inequalities between norms of partial derivatives on convex domains

We consider inequalities between $L_p$-norms of partial derivatives, $p\in [1,+\infty]$, for bivariate concave functions on a convex domain that vanish on the boundary. Can the ratio between those norms be arbitrarily large? If not, what is the upper bound? We show that for $p=1$, the ratio is always bounded and find sharp estimates, while for $p>1$, the answer depends on the geometry of the domain.

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arXiv2024

The problem of optimal camouflaging

We consider the problem of camouflaging for bodies with specular surface in the framework of geometric optics. The index of visibility introduced in [Plakhov 2017] measures the mean deviation of light rays incident on the body's surface. We study the problem of minimal visibility index for bodies with fixed volume contained in the unit sphere. This problem is reduced to a special vector-valued problem of optimal mass transfer, which is solved partly analytically and partly numerically. This paper is a continuation of the study started in [Plakhov 2009], [Plakhov 2017], and [Plakhov and Roshchina 2011].

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arXiv2024

Local minima in Newton's aerodynamical problem and inequalities between norms of partial derivatives

The problem considered first by I. Newton (1687) consists in finding a surface of the minimal frontal resistance in a parallel flow of non-interacting point particles. The standard formulation assumes that the surface is convex with a given convex base $Ω$ and a bounded altitude. Newton found the solution for surfaces of revolution. Without this assumption the problem is still unsolved, although many important results have been obtained in the last decades. We consider the problem to characterize the domains $Ω$ for which the flat surface gives a local minimum. We show that this problem can be reduced to an inequality between $L_2$-norms of partial derivatives for bivariate concave functions on a convex domain that vanish on the boundary. Can the ratio between those norms be arbitrarily large? The answer depends on the geometry of the domain. A complete criterion is derived, which also solves the local minimality problem.

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arXiv2023

Rotating rod and ball

We consider a mechanical system consisting of an infinite rod (a straight line) and a ball (a massless point) on the plane. The rod rotates uniformly around one of its points. The ball is reflected elastically when colliding with the rod and moves freely between consecutive hits. A sliding motion along the rod is also allowed. We prove the existence and uniqueness of the motion with a given position and velocity at a certain time instant. We prove that only 5 kinds of motion are possible: a billiard motion; a sliding motion; a billiard motion followed by sliding; a sliding motion followed by a billiard one; and a constant motion when the ball is at the center of rotation. The asymptotic behaviors of time intervals between consecutive hits and of distances between the points of hits on the rod are determined.

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arXiv2022

Local structure of convex surfaces near regular and conical points

Consider a point on a convex surface in $\mathbb{R}^d$, $d \ge 2$ and a plane of support $Π$ to the surface at this point. Draw a plane parallel to $Π$ cutting a part of the surface. We study the limiting behavior of this part of surface when the plane approaches the point, being always parallel to $Π$. More precisely, we study the limiting behavior of the normalized surface area measure in $S^{d-1}$ induced by this part of surface. In this paper we consider two cases: (a) when the point is regular and (b) when it is singular conical, that is, the tangent cone at the point does not contain straight lines. In the case (a) the limit is the atom located at the outward normal vector to $Π$, and in the case (b) the limit is equal to the measure induced by the part of the tangent cone cut off by a plane.

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arXiv2022

On the structure of singular points of a solution to Newton's least resistance problem

We consider the following problem stated in 1993 by Buttazzo and Kawohl: minimize the functional $\int\!\!\int_Ω(1 + |\nabla u(x,y)|^2)^{-1} dx\, dy$ in the class of concave functions $u: Ω\to [0,M]$, where $Ω\subset \mathbb{R}^2$ is a convex domain and $M > 0$. It generalizes the classical minimization problem, which was initially stated by I. Newton in 1687 in the more restricted class of radial functions. The problem is not solved until now; there is even nothing known about the structure of singular points of a solution. In this paper we, first, solve a family of auxiliary 2D least resistance problems and, second, apply the obtained results to study singular points of a solution to our original problem. More precisely, we derive a necessary condition for a point being a ridge singular point of a solution and prove, in particular, that all ridge singular points with horizontal edge lie on the top level and zero level sets.

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arXiv2022

A solution to Newton's least resistance problem is uniquely defined by its singular set

Let $u$ minimize the functional $F(u) = \int_Ωf(\nabla u(x))\, dx$ in the class of convex functions $u : Ω\to {\mathbb R}$ satisfying $0 \le u \le M$, where $Ω\subset {\mathbb R}^2$ is a compact convex domain with nonempty interior and $M > 0$, and $f : {\mathbb R}^2 \to {\mathbb R}$ is a $C^2$ function, with $\{ ξ: \, \text{the smallest eigenvalue of} \, f"(ξ) \, \text{is zero} \}$ being a closed nowhere dense set in ${\mathbb R}^2$. Let epi$(u)$ denote the epigraph of $u$. Then any extremal point $(x, u(x))$ of epi$(u)$ is contained in the closure of the set of singular points of epi$(u)$. As a consequence, an optimal function $u$ is uniquely defined by the set of singular points of epi$(u)$. This result is applicable to the classical Newton's problem, where $F(u) = \int_Ω(1 + |\nabla u(x)|^2)^{-1}\, dx$.

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arXiv2020

Method of nose stretching in Newton's problem of minimal resistance

We consider the problem $\inf\big\{ \int\!\!\int_Ω(1 + |\nabla u(x,y)|^2)^{-1} dx dy : \text{ the function } u : Ω\to \mathbb{R} \text{ is concave and } 0 \le u(x,y) \le M \text{ for all } (x,y) \in Ω=\{ (x,y): x^2 + y^2 \le 1 \} \, \big\}$ (Newton's problem) and its generalizations. In the paper \cite{BrFK} it is proved that if a solution $u$ is $C^2$ in an open set $\mathcal{U} \subset Ω$ then $\det D^2u = 0$ in $\mathcal{U}$. It follows that graph$(u)\rfloor_\mathcal{U}$ does not contain extreme points of the subgraph of $u$. In this paper we prove a somewhat stronger result. Namely, there exists a solution $u$ possessing the following property. If $u$ is $C^1$ in an open set $\mathcal{U} \subset Ω$ then graph$(u\rfloor_\mathcal{U})$ does not contain extreme points of the convex body $C_u = \{ (x,y,z) :\, (x,y) \in Ω,\ 0 \le z \le u(x,y) \}$. As a consequence, we have $C_u = \text{\rm Conv} (\overline{\text{\rm Sing$C_u$}})$, where Sing$C_u$ denotes the set of singular points of $\partial C_u$. We prove a similar result for a generalized Newton's problem.

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arXiv2019

A note on Newton's problem of minimal resistance for convex bodies

We consider the following problem: minimize the functional $\int_Ωf(\nabla u(x))\, dx$ in the class of concave functions $u: Ω\to [0,M]$, where $Ω\subset \mathbb{R}^2$ is a convex body and $M > 0$. If $f(x) = 1/(1 + |x|^2)$ and $Ω$ is a circle, the problem is called Newton's problem of least resistance. It is known that the problem admits at least one solution. We prove that if all points of $\partialΩ$ are regular and ${|x|f(x)}/(|y|f(y)) \to +\infty$ as $|x|/|y| \to 0$ then a solution $u$ to the problem satisfies $u\rfloor_{\partialΩ} = 0$. This result proves the conjecture stated in 1993 for Newton's problem.

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arXiv2019

Behavior of convex surfaces near ridge points

The aim of this paper is twofold. First, we cut off a part of a convex surface by a plane near a ridge point and characterize the limiting behavior of the surface measure in $S^2$ induced by this part of surface when the plane approaches the point. Second, this characterization is applied to Newton's least resistance problem for convex bodies: minimize the functional $\int\int_Ω(1 + |\nabla u(x,y)|^2)^{-1} dx dy$ in the class of convex functions $u: Ω\to [0,M]$, where $Ω\subset R^2$ is a convex body and $M > 0$. It has been known that if $u_*$ solves the problem then $|\nabla u_*(x,y)| \ge 1$ at all regular points $(x,y)$ such that $u_*(x,y) > 0$. We prove that if the lower level set $L_0 = \{ (x,y): u_*(x,y) = 0 \}$ has nonempty interior, then for almost all points of its boundary $(\bar x, \bar y) \in \partial L_0$ one has $\lim_{\stackrel{(x,y)\to(\bar x,\bar y)}{u_*(x,y)>0}}|\nabla u_*(x,y)| = 1$.

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arXiv2018

Local properties of the surface measure of convex bodies

It is well known that any measure in S^2 satisfying certain simple conditions is the surface measure of a bounded convex body in R^3. It is also known that a local perturbation of the surface measure may lead to a nonlocal perturbation of the corresponding convex body. We prove that, under mild conditions on a convex body, there are families of perturbations of its surface measure forming line segments, with the original measure at the midpoint, leading to local perturbations of the body. Moreover, there is, in a sense, a huge amount of such families. We apply this result to Newton's problem of minimal resistance for convex bodies.

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arXiv2018

Optimal Impulse Control of Dynamical Systems

Using the tools of the Markov Decision Processes, we justify the dynamic programming approach to the optimal impulse control of deterministic dynamical systems. We prove the equivalence of the integral and differential forms of the optimality equation. The theory is illustrated by an example from mathematical epidemiology. The developed methods can be also useful for the study of piecewise deterministic Markov processes.

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arXiv2017

The problem of camouflaging via mirror reflections

This work is related to billiards and their applications in geometric optics. It is known that perfectly invisible bodies with mirror surface do not exist. It is natural to search for bodies that are, in a sense, close to invisible. We introduce a {\it visibility index} of a body measuring the mean angle of deviation of incident light rays, and derive a lower estimate to this index. This estimate is a function of the body's volume and of the minimal radius of a ball containing the body. This result is far from being final and opens a possibility for further research.

math.MG↗
arXiv2015

Plane sets invisible in finitely many directions

We consider the problem of mirror invisibility for plane sets. Given a circle and a finite number of unit vectors (defining the directions of invisibility) such that the angles between them are commensurable with $π$, for any $\varepsilon > 0$ there exists a set invisible in the chosen directions that contains the circle and is contained in its $\varepsilon$-neighborhood. This set is the disjoint union of infinitely many domains with piecewise smooth boundary.

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arXiv2014

Minimal resistance of curves under the single impact assumption

We consider the hollow on the half-plane $\{(x,y) : y \le 0\} \subset \mathbb{R}^2$ defined by a function $u : (-1, 1) \to \mathbb{R}$, $u(x) < 0$ and a vertical flow of point particles incident on the hollow. It is assumed that $u$ satisfies the so-called single impact condition (SIC): each incident particle is elastically reflected by graph$(u)$ and goes away without hitting the graph of $u$ anymore. We solve the problem: find the function $u$ minimizing the force of resistance created by the flow. We show that the graph of the minimizer is formed by two arcs of parabolas symmetric to each other with respect to the $y$-axis. Assuming that the resistance of $u \equiv 0$ equals 1, we show that the minimal resistance equals $π/2 - 2\arctan(1/2) \approx 0.6435$. This result completes the previously obtained result stating in particular that the minimal resistance of a hollow in higher dimensions equals 0.5. We additionally consider a similar problem of minimal resistance, where the hollow in the half-space $\{(x_1,\ldots,x_d, y) : y \le 0 \} \subset \mathbb{R}^{d+1}$ is defined by a radial function $U$ satisfying SIC, $U(x) = u(|x|)$, with $x = (x_1,\ldots,x_d), u(ξ) < 0$ for $0 \le ξ< 1$ and $u(ξ) = 0$ for $ξ\ge 1$, and the flow is parallel to the $y$-axis. The minimal resistance is greater than $0.5$ (and coincides with $0.6435$ when $d = 1$) and converges to $0.5$ as $d \to \infty$.

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arXiv2014

Newton's problem of minimal resistance under the single-impact assumption

A parallel flow of non-interacting point particles is incident on a body at rest. When hitting the body's surface, the particles are reflected elastically. Assume that each particle hits the body at most once (SIC condition); then the force of resistance of the body along the flow direction can be written down in a simple analytical form. The problem of minimal resistance within this model was first considered by Newton (1687) in the class of bodies with a fixed length M along the flow direction and with a fixed maximum orthogonal cross section, under the additional conditions that the body is convex and rotationally symmetric. Here we solve the problem (first stated by Ferone, Buttazzo, and Kawohl in 1995) for the wider class of bodies satisfying SIC and with the additional conditions removed. The scheme of solution is inspired by Besicovitch's method of solving the Kakeya problem. If the maximum cross section is a disc, the decrease of resistance as compared with the original Newton problem is more than twofold; the ratio tends to 2 as M goes to 0 and to 81/4 as M goes to infinity. We also prove that the infimum of resistance is 0 for a wider class of bodies with both single and double impacts allowed.

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