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

A. A. Uspenskii

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 250-263

Résumé

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.

Détail

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

Bibliographie

[1] Kruzhkov S.N., “Obobschennye resheniya uravnenii Gamiltona – Yakobi tipa eikonala. I”, Mat. sb., 98:3 (1974), 450–493

[2] Subbotin A.I., “Obobschenie osnovnogo uravneniya teorii differentsialnykh igr”, Dokl. AN SSSR, 254:2 (1980), 293–297 | MR | Zbl

[3] Crandall M.G., Lions P.L., Viscosity solutions of Hamilton-Jacobi equations, 277:1 (1983), 1–42 | MR | Zbl

[4] Subbotin A.I., Obobschennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka. Perspektivy dinamicheskoi optimizatsii, Institut kompyuternykh tekhnologii, M.; Izhevsk, 2003, 336 pp.

[5] Kolokoltsov V.N., Maslov V.P., “Zadacha Koshi dlya odnorodnogo uravneniya Bellmana”, Dokl. AN SSSR, 296:4 (1987), 796–800 | MR

[6] Krasovskii N.N., Subbotin A.I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp. | MR

[7] Uspenskii A.A., Ushakov V.N., Fomin A.N., $\alpha$-mnozhestva i ikh svoistva, In-t matematiki i mekhaniki UrO RAN, Ekaterinburg, 2004, 62 pp.

[8] Uspenskii A.A., Analiticheskie metody vychisleniya mery nevypuklosti ploskikh mnozhestv, In-t matematiki i mekhaniki UrO RAN, Ekaterinburg, 2007, 21 pp.

[9] Uspenskii A.A., Lebedev P.D., “Issledovanie geometrii i asimptotiki volnovykh frontov v nekotorykh zadachakh upravleniya”, Tr. 9-i Mezhdunar. Chetaevskoi konf., v. 5, 2007, 224–236

[10] Uspenskii A.A., Lebedev P.D., “Geometriya i asimptotika volnovykh frontov”, Izv. vuzov, 2008, no. 3, 27–37 | MR | Zbl

[11] Ushakov V.N., Uspenskii A.A., Lebedev P.D., “Postroenie minimaksnogo resheniya uravneniya tipa eikonala”, Tr. In-ta matematiki i mekhaniki UrO RAN, 14:2 (2008), 182–191 | MR | Zbl

[12] Uspenskii A.A., Lebedev P.D., “Usloviya transversalnosti vetvei resheniya nelineinogo uravneniya v zadache bystrodeistviya s krugovoi indikatrisoi”, Tr. In-ta matematiki i mekhaniki UrO RAN, 14:4 (2008), 82–100 | MR

[13] Uspenskii A.A., Lebedev P.D., “Postroenie funktsii optimalnogo rezultata v zadache bystrodeistviya na osnove mnozhestva simmetrii”, Avtomatika i telemekhanika, 2009, no. 7, 50–57 | MR | Zbl

[14] Uspenskii A.A., Lebedev P.D., “O mnozhestve predelnykh znachenii lokalnykh diffeomorfizmov pri evolyutsii volnovykh frontov”, Tr. In-ta matematiki i mekhaniki UrO RAN, 16:1 (2010), 171–186 | MR

[15] Ushakov V.N., Uspenskii A.A., Lebedev P.D., “Geometriya singulyarnykh krivykh dlya odnogo klassa zadach bystrodeistviya”, Vestn. S.-Peterb. un-ta Ser. 10, 2013, no. 3, 157–167

[16] Uspenskii A.A., “Formuly ischisleniya negladkikh osobennostei funktsii optimalnogo rezultata v zadache bystrodeistviya”, Tr. In-ta matematiki i mekhaniki UrO RAN, 20:3 (2014), 276–290

[17] Slyusarev G.G., Geometricheskaya optika, Izd-vo AN SSSR, M.-L., 1946, 332 pp. | MR

[18] Byushgens S.S., Differentsialnaya geometriya, GITTL, M., 1940, 300 pp.

[19] Arnold V.I., Osobennosti kaustik i volnovykh frontov, Fazis, M., 1996, 334 pp. | MR

[20] Brus Dzh., Dzhiblin P., Krivye i osobennosti, Mir, M., 1988, 262 pp. | MR

[21] Zakalyukin V.M., “Ogibayuschie semeistv volnovykh frontov i teoriya upravleniya”, Tr. MIAN, 209 (1995), 133–142 | MR | Zbl

[22] Blyashke V., Krug i shar, Nauka, M., 1967, 232 pp. | MR

[23] Natanson I.P., Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974, 480 pp. | MR

[24] Subbotina N.N., Kolpakova E.A., “O strukture lokalno lipshitsevykh minimaksnykh reshenii uravneniya Gamiltona - Yakobi - Bellmana v terminakh klassicheskikh kharakteristik”, Tr. In-ta matematiki i mekhaniki UrO RAN, 15:3 (2009), 202–218

[25] Ohm Martin, Lehrbuch der gesamten hohern Mathematik, Bd 2, Friedrich Volckmar, Leipzig, 1835