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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2018, Volume 28, Issue 1, Pages 59–73 (Mi vuu620)

MATHEMATICS

Identification of the singularity of the generalized solution of the Dirichlet problem for an eikonal type equation under the conditions of minimal smoothness of a boundary set

A. A. Uspenskii, P. D. Lebedev

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16, Yekaterinburg, 620219, Russia

Abstract: The subject of the study is pseudo-vertices of a boundary set, which are necessary for the analytical and numerical construction of singular branches of the generalized (minimax) solution of the Dirichlet problem for an eikonal type equation. The case of variable smoothness of the boundary set boundary is considered, under which the order of smoothness at the points of consideration is reduced to the lowest possible value - up to one. Necessary conditions for the existence of pseudo-vertices are obtained, expressed in terms of one-sided partial limits of differential relations, depending on the properties of local diffeomorphisms that determine these points. An example is given that illustrates the application of the results obtained while solving the velocity problem.

Keywords: first-order partial differential equation, minimax solution, velocity, wave front, diffeomorphism, eikonal, optimal result function, singular set, symmetry, pseudo-vertex.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-01-00264

DOI: https://doi.org/10.20537/vm180106

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Bibliographic databases:

UDC: 517.977
MSC: 35A18

Citation: A. A. Uspenskii, P. D. Lebedev, “Identification of the singularity of the generalized solution of the Dirichlet problem for an eikonal type equation under the conditions of minimal smoothness of a boundary set”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 28:1 (2018), 59–73

Citation in format AMSBIB
\Bibitem{UspLeb18} \by A.~A.~Uspenskii, P.~D.~Lebedev \paper Identification of the singularity of the generalized solution of the Dirichlet problem for an eikonal type equation under the conditions of minimal smoothness of a boundary set \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2018 \vol 28 \issue 1 \pages 59--73 \mathnet{http://mi.mathnet.ru/vuu620} \crossref{https://doi.org/10.20537/vm180106} \elib{http://elibrary.ru/item.asp?id=32697216} 

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This publication is cited in the following articles:
1. Rodin A.S., Shagalova L.G., “Bifurcation Points of the Generalized Solution of the Hamilton-Jacobi-Bellman Equation”, IFAC PAPERSONLINE, 51:32 (2018), 866–870
2. P. D. Lebedev, A. A. Uspenskii, “Postroenie resheniya zadachi upravleniya po bystrodeistviyu pri narushenii gladkosti krivizny granitsy tselevogo mnozhestva”, Izv. IMI UdGU, 53 (2019), 98–114
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