The Geometry of a Singular Set of Hypersurfaces and the Eikonal Equation
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Geometry of a Singular Set of Hypersurfaces and the Eikonal Equation I. G. Tsar’kov1, 2* 1
2
Lomonosov Moscow State University, Moscow, 119991 Russia Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991 Russia Received January 2, 2020; in final form, March 30, 2020; accepted April 15, 2020
Abstract—Smooth solutions of the eikonal equation are studied. A relationship between the geometry of a hypersurface and the set of singular points of its metric function on both sides of this hypersurface is investigated. DOI: 10.1134/S0001434620090114 Keywords: eikonal equation, singular set, regular point, sun point.
1. INTRODUCTION Let (x, M ) denote the quantity inf y∈M x − y equal to the distance from a point x ∈ X to a set M in a Banach space X. The function ( · , M ) is often called the metric function. A point y0 ∈ M is said to be nearest to x in M if x − y0 = (x, M ). The set of points nearest to x in M is denoted by P x = PM x. By B(x, r) we denote the ball of radius r ≥ 0 centered at x, i.e., the set {y ∈ X | y − x ≤ r}. This paper is devoted to one of the new promising directions of geometric approximation theory, which studies, by its own methods, solutions of the Hamilton–Jacobi equations, one of which is the eikonal equation ∇uX ∗ = 1. Here X ∗ is the space dual to X and u = u(x) is a function on a subset of X. This equation is a natural generalization of the classical eikonal equation |∇u| = 1, where the function u = u(x) (a solution of this equation) is usually given on an open set Ω ⊂ Rn . It easily follows from simplest considerations that such solutions are locally functions of the form c ± (x, M ) for some constants c ∈ R and subsets M ⊂ X. The eikonal equation itself is the main equation of geometric optics, whose foundations are exposed in the monographs [1] by Born and Wolf, [2] by Bruce and Giblin, [3] by Feynman, [4] by Kravtsov and Orlov, [5] by Arnold, and [6] by Kruzhkov. Here we take notice of the following fact: the smoothness of a solution of the eikonal equation is related to the geometric-approximation notion of the local solarity of the level surfaces of this solution. For this reason, in this paper, we shall investigate the local solarity (regularity) properties and singular points of hypersurfaces. We note that various versions of the notion of solarity have arisen in approximation theory out of touch with geometric optics. For the first time, this notion was defined by N. V. Efimov and S. B. Stechkin, who noticed that, from a Chebyshev compact set (i.e., a set for which nearest points exist and are unique), rays emanate such that the starting point of each ray is nearest to all points of this ray. In what follows, it had become clear that this solarity property is a geometric reformulation of Kolmogorov’s criterion (a characterization of a best approximation element). In an incomprehensible mysterious way, the names “sun” and “rays” have turned out to be related to basic notions of geometric optics, such as emitting surface, wave front, and light ray. More details
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