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 Izv. RAN. Ser. Mat., 2019, Volume 83, Issue 6, Pages 167–194 (Mi izv8850)

Smooth solutions of the eikonal equation and the behaviour of local minima of the distance function

I. G. Tsar'kov

Lomonosov Moscow State University

Abstract: We study smooth solutions of the eikonal equation. To do this, we investigate the problem of geometric-topological properties of the singularities of the distance function and the regular set. We establish a connection between the caustic and domains where the number of local minima of the distance function is constant. We pose a number of problems about reflecting surfaces bringing light to a single point (a focus) and introduce the notions of generalized ellipsoids and paraboloids.

Keywords: singular sets, regular sets, solarity points, eikonal equation, caustic.

 Funding Agency Grant Number Russian Foundation for Basic Research 19-01-00332-a This paper has been written with the support of the Russian Foundation for Basic Research (grant no. 19-01-00332-a).

DOI: https://doi.org/10.4213/im8850

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English version:
Izvestiya: Mathematics, 2019, 83:6, 1234–1258

Bibliographic databases:

UDC: 517.9
MSC: Primary 41A65; Secondary 54C60, 78A05

Citation: I. G. Tsar'kov, “Smooth solutions of the eikonal equation and the behaviour of local minima of the distance function”, Izv. RAN. Ser. Mat., 83:6 (2019), 167–194; Izv. Math., 83:6 (2019), 1234–1258

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv8850
• https://doi.org/10.4213/im8850
• http://mi.mathnet.ru/eng/izv/v83/i6/p167

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. R. Alimov, “Vypuklost i monotonnaya lineinaya svyaznost mnozhestv s nepreryvnoi metricheskoi proektsiei v trekhmernykh prostranstvakh”, Tr. IMM UrO RAN, 26, no. 2, 2020, 28–46
2. I. G. Tsar'kov, “The Geometry of a Singular Set of Hypersurfaces and the Eikonal Equation”, Math. Notes, 108:3 (2020), 426–433
3. A. R. Alimov, “Characterization of Sets with Continuous Metric Projection in the Space $\ell^\infty_n$”, Math. Notes, 108:3 (2020), 309–317
4. S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. I. Shafarevich, “Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations”, Russian Math. Surveys, 76:5 (2021), 745–819
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