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## Archive of Issues

Russia Yekaterinburg
Year
2018
Volume
28
Issue
1
Pages
59-73
 Section Mathematics Title 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 Author(-s) Uspenskii A.A.a, Lebedev P.D.a Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa 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 UDC 517.977 MSC 35A18 DOI 10.20537/vm180106 Received 1 February 2018 Language Russian Citation Uspenskii A.A., Lebedev P.D. 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, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 1, pp. 59-73. References Kruzhkov S.N. Generalized solutions of the Hamilton-Jacobi equations of eikonal type. I. Formulation of the problems; existence, uniqueness and stability theorems; some properties of the solutions, Mathematics of the USSR-Sbornik, 1975, vol. 27, no. 3, pp. 406-446. DOI: 10.1070/SM1975v027n03ABEH002522 Crandall M.G., Lions P.-L. Viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 1983, vol. 277, no. 1, pp. 1-42. DOI: 10.1090/S0002-9947-1983-0690039-8 Subbotin A.I. Generalized solutions of first order PDEs: the dynamical optimization perspective, Boston: Birkhäuser, 1995, XII+314 p. DOI: 10.1007/978-1-4612-0847-1 Krasovskii N.N., Subbotin A.I. Pozitsionnye differentsial'nye igry (Positional differential games), Moscow: Nauka, 1974, 456 p. Grigor’eva S.V., Pakhotinskikh V.Yu., Uspenskii A.A., Ushakov V.N. Construction of solutions in certain differential games with phase constraints, Sbornik: Mathematics, 2005, vol. 196, no. 4, pp. 513-539. DOI: 10.1070/SM2005v196n04ABEH000890 Lebedev P.D., Uspenskii A.A., Ushakov V.N. Construction of a minimax solution for an eikonal-type equation, Proceedings of the Steklov Institute of Mathematics, 2008, vol. 263, suppl. 2, pp. S191-S201. DOI: 10.1134/S0081543808060175 Uspenskii A.A., Lebedev P.D. Construction of the optimal outcome function for a time-optimal problem on the basis of a symmetry set, Automation and Remote Control, 2009, vol. 70, no. 7, pp. 1132-1139. DOI: 10.1134/S0005117909070054 Uspenskii A.A., Lebedev P.D. On the set of limit values of local diffeomorphisms in wavefront evolution, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 272, suppl. 1, pp. S255-S270. DOI: 10.1134/S0081543811020180 Uspenskii A.A. Calculation formulas for nonsmooth singularities of the optimal result function in a time-optimal problem, Proceedings of the Steklov Institute of Mathematics, 2015, vol. 291, suppl. 1, pp. S239-S254. DOI: 10.1134/S0081543815090163 Uspenskii A.A., Lebedev P.D. The construction of singular curves for generalized solutions of eikonal-type equations with a curvature break in the boundary of the edge set, Proceedings of the Steklov Institute of Mathematics, 2017, vol. 297, suppl. 1, pp. S191-S202. DOI: 10.1134/S0081543817050212 Uspenskii A.A. Necessary conditions for the existence of pseudovertices of the boundary set in the Dirichlet problem for the eikonal equation, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2015, vol. 21, no. 1, pp. 250-263 (in Russian). Arnold V.I. Singularities of caustics and wave fronts, Springer Netherlands, 1990, XIII+259 p. DOI: 10.1007/978-94-011-3330-2 Zakalyukin V.M., Kurbatskii A.N. Envelope singularities of families of planes in control theory, Proceedings of the Steklov Institute of Mathematics, 2008, vol. 262. issue 1, pp. 66-79. DOI: 10.1134/S0081543808030073 Borovskikh A.V. Equivalence groups of eikonal equation and classes of equivalent equations, Vestnik Novosibirskogo Gosudarstvennogo Universiteta, 2006, no. 4, pp. 3-42 (in Russian). Sethian J.A., Vladimirsky A. Fast methods for the Eikonal and related Hamilton-Jacobi equations on unstructured meshes, Proc. Natl. Acad. Sci. USA, 2000, vol. 97, no. 11, pp. 5699-5703. DOI: 10.1073/pnas.090060097 Uspenskii A.A., Ushakov V.N., Fomin A.N. $\alpha$-sets and their properties, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 2004, 62 p. Deposited in VINITI 02.04.2004, no. 543-В2004 (in Russian). Ushakov V.N., Uspenskii A.A. $\alpha$-sets in finite dimensional Euclidean spaces and their properties, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 1, pp. 95-120. DOI: 10.20537/vm160109 Bruce J.W., Giblin P.J. Curves and singularities, Cambridge: Cambridge University Press, 1984, 222 p. Translated under the title Krivye i osobennosti, Moscow: Mir, 1988, 262 p. Uspenskii A.A. Derivatives with respect to diffeomorphisms and their applications in control theory and geometrical optics, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 293, suppl. 1, pp. 238-253. DOI: 10.1134/S0081543816050217 Ohm M. Lehrbuch der gesamten höhern Mathematik. Bd. 2., Leipzig: Verlag Friedrich Volckmar, 1835. Full text