A. A. Uspenskii, “Necessary conditions for the existence of pseudovertices of the boundary set in the Dirichlet problem for the eikonal equation”, Trudy Inst. Mat. i Mekh. UrO RAN, 21, no. 1, 2015, 250

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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2015, Volume 21, Number 1, Pages 250–263 (Mi timm1162)

This article is cited in 12 scientific papers (total in 12 papers)

Necessary conditions for the existence of pseudovertices of the boundary set in the Dirichlet problem for the eikonal equation

A. A. Uspenskii

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

Full-text PDF (785 kB) Citations (12)

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Abstract: The problem of the appearance of nonsmooth singularities in generalized solutions of first-order PDEs is studied. The Dirichlet boundary value problem is considered for an eikonal-type equation. The subject of the research is pseudovertices of the boundary set. Pseudovertices are useful for the analytic and numerical construction of branches of the singular set, i.e., the set where the solution of the boundary value problem is nonsmooth. Necessary conditions for the existence of pseudovertices are obtained in the case when a nonconvex boundary set has smooth boundary. The conditions are written in terms of constant curvature and constant coordinate functions defining the boundary of the set.

Keywords: first-order PDE; minimax solution; wavefront; diffeomorphism; eikonal; optimal result function; singular set; symmetry.

Received: 10.12.2014

Bibliographic databases:

Document Type: Article

UDC: 517.954

Language: Russian

Citation: A. A. Uspenskii, “Necessary conditions for the existence of pseudovertices of the boundary set in the Dirichlet problem for the eikonal equation”, Trudy Inst. Mat. i Mekh. UrO RAN, 21, no. 1, 2015, 250–263

Citation in format AMSBIB

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\by A.~A.~Uspenskii

\paper Necessary conditions for the existence of pseudovertices of the boundary set in the Dirichlet problem for the eikonal equation

\serial Trudy Inst. Mat. i Mekh. UrO RAN

\yr 2015

\vol 21

\issue 1

\pages 250--263

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\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3407899}

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Linking options:

https://www.mathnet.ru/eng/timm1162

https://www.mathnet.ru/eng/timm/v21/i1/p250

This publication is cited in the following 12 articles:

P. D. Lebedev, A. A. Uspenskii, “Metod Nyutona pri postroenii singulyarnogo mnozhestva minimaksnogo resheniya v odnom klasse kraevykh zadach dlya uravnenii Gamiltona — Yakobi”, Chelyab. fiz.-matem. zhurn., 9:1 (2024), 63–76
P. D. Lebedev, A. A. Uspenskii, “Numerical-analytic construction of a generalized solution to the eikonal equation in the plane case”, Sb. Math., 215:9 (2024), 1224–1248
A. A. Uspenskii, P. D. Lebedev, “On singularity structure of minimax solution to Dirichlet problem for eikonal type equation with discontinuous curvature of boundary of boundary set”, Ufa Math. J., 13:3 (2021), 126–151
P. D. Lebedev, A. A. Uspenskii, “Postroenie rasseivayuschikh krivykh v odnom klasse zadach bystrodeistviya pri skachkakh krivizny granitsy tselevogo mnozhestva”, Izv. IMI UdGU, 55 (2020), 93–112
P. D. Lebedev, A. A. Uspenskii, “Elementy analiticheskogo konstruktora reshenii v klasse zadach upravleniya po bystrodeistviyu s tselevym mnozhestvom s razryvnoi kriviznoi granitsy”, Vestnik rossiiskikh universitetov. Matematika, 25:132 (2020), 370–386
A. A. Uspenskii, P. D. Lebedev, “Svoistva nestatsionarnykh psevdovershin kraevogo mnozhestva pri razryve gladkosti krivizny ego granitsy v zadache Dirikhle dlya uravneniya tipa eikonala”, Sib. elektron. matem. izv., 17 (2020), 2028–2044
A. A. Uspenskii, P. D. Lebedev, “Vyyavlenie singulyarnosti u obobschennogo resheniya zadachi Dirikhle dlya uravneniya tipa eikonala v usloviyakh minimalnoi gladkosti granitsy kraevogo mnozhestva”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:1 (2018), 59–73
V. N. Ushakov, A. A. Uspenskii, A. A. Ershov, “Alfa-mnozhestva v konechnomernykh evklidovykh prostranstvakh i ikh prilozheniya v teorii upravleniya”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 14:3 (2018), 261–272
P. D. Lebedev, A. A. Uspenskii, “Construction of singular sets in a velocity control problem with nonconvex target”, IFAC-PapersOnLine, 51:32 (2018), 681–686
A. A. Uspenskii, P. D. Lebedev, “Evklidovo rasstoyanie do zamknutogo mnozhestva kak minimaksnoe reshenie zadachi Dirikhle dlya uravneniya Gamiltona-Yakobi”, Vestnik Tambovskogo universiteta. Seriya: estestvennye i tekhnicheskie nauki, 23:124 (2018), 797–804
A. A. Uspenskii, P. D. Lebedev, “The construction of singular curves for generalized solutions of eikonal-type equations with a curvature break in the boundary of the boundary set”, Proc. Steklov Inst. Math. (Suppl.), 297, suppl. 1 (2017), 191–202
P. D. Lebedev, A. A. Uspenskii, “Postroenie funktsii optimalnogo rezultata i rasseivayuschikh linii v zadachakh bystrodeistviya s nevypuklym tselevym mnozhestvom”, Tr. IMM UrO RAN, 22, no. 2, 2016, 188–198

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Trudy Instituta Matematiki i Mekhaniki UrO RAN

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