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St. Petersburg Mathematical Journal

This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.

ISSN 1547-7371 (online) ISSN 1061-0022 (print)

The 2020 MCQ for St. Petersburg Mathematical Journal is 0.68.

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Solution of the Hadamard problem in the class of stepwise gauge-equivalent deformations of homogeneous differential operators with constant coefficients
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by S. P. Khekalo
Translated by: the author
St. Petersburg Math. J. 19 (2008), 1015-1028
DOI: https://doi.org/10.1090/S1061-0022-08-01034-0
Published electronically: August 22, 2008
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Abstract:

In the paper, all nontrivial Huygens stepwise gauge-equivalent deformations for a priori Huygens homogeneous differential operators with constant coefficients are described explicitly. A condition is obtained under which an operator in the class of stepwise gauge-equivalent operators is Huygens, and new examples are given of iso-Huygens deformations of radial homogeneous differential operators of higher order.
References
  • J. Hadamard, Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques, Hermann, Paris, 1932.
  • N. Kh. Ibragimov, Gruppy preobrazovaniĭ v matematicheskoĭ fizike, “Nauka”, Moscow, 1983 (Russian). MR 734307
  • Yu. Yu. Berest and A. P. Veselov, The Huygens principle and integrability, Uspekhi Mat. Nauk 49 (1994), no. 6(300), 7–78 (Russian); English transl., Russian Math. Surveys 49 (1994), no. 6, 5–77. MR 1316866, DOI 10.1070/RM1994v049n06ABEH002447
  • Karl L. Stellmacher, Ein Beispiel einer Huyghensschen Differentialgleichung, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt. 1953 (1953), 133–138 (German). MR 60695
  • J. E. Lagnese and K. L. Stellmacher, A method of generating classes of Huygens’ operators, J. Math. Mech. 17 (1967), 461–472. MR 0217409, DOI 10.1512/iumj.1968.17.17027
  • G. Darboux, Sur la représentation sphérique des surfaces, Compt. Rend. (Paris) 94 (1882), 1343–1345.
  • Yuri Berest and Yuri Molchanov, Fundamental solutions for partial differential equations with reflection group invariance, J. Math. Phys. 36 (1995), no. 8, 4324–4339. MR 1341994, DOI 10.1063/1.530964
  • Yuri Berest, Hierarchies of Huygens’ operators and Hadamard’s conjecture, Acta Appl. Math. 53 (1998), no. 2, 125–185. MR 1646583, DOI 10.1023/A:1006069012474
  • S. P. Khèkalo, Iso-Huygens deformations of the Cayley operator by the general Lagnese-Stellmacher potential, Izv. Ross. Akad. Nauk Ser. Mat. 67 (2003), no. 4, 189–212 (Russian, with Russian summary); English transl., Izv. Math. 67 (2003), no. 4, 815–836. MR 2001768, DOI 10.1070/IM2003v067n04ABEH000446
  • Yuri Berest, The problem of lacunas and analysis on root systems, Trans. Amer. Math. Soc. 352 (2000), no. 8, 3743–3776. MR 1694280, DOI 10.1090/S0002-9947-00-02543-5
  • I. G. Petrovskiĭ, Izbrannye trudy, “Nauka”, Moscow, 1986 (Russian). Sistemy uravneniĭ s chastnymi proizvodnymi. Algebraicheskaya geometriya. [Systems of partial differential equations. Algebraic geometry]; Edited and with a preface by V. I. Arnol′d, N. N. Bogolyubov, A. N. Kolmogorov, O. A. Oleĭnik, S. L. Sobolev and A. N. Tikhonov; Compiled by Oleĭnik; With commentaries by Kolmogorov, L. R. Volevich, V. Ya. Ivriĭ, I. M. Gel′fand, G. E. Shilov, Oleĭnik, V. P. Palamodov, A. M. Gabrièlov and V. M. Kharlamov. MR 871873
  • S. G. Gindikin, The Cauchy problem for strongly homogeneous differential operators, Trudy Moskov. Mat. Obšč. 16 (1967), 181–208 (Russian). MR 0227593
  • V. M. Babich, Hadamard’s ansatz, its analogues, generalizations and applications, Algebra i Analiz 3 (1991), no. 5, 1–37 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 3 (1992), no. 5, 937–972. MR 1186234
  • Paul Günther, Ein Beispiel einer nichttrivialen Huygensschen Differentialgleichung mit vier unabhängigen Variablen, Arch. Rational Mech. Anal. 18 (1965), 103–106 (German). MR 174865, DOI 10.1007/BF00282256
  • M. A. Semenov-Tjan-Šanskiĭ, Harmonic analysis on Riemannian symmetric spaces of negative curvature, and scattering theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 3, 562–592, 710 (Russian). MR 0467179
  • Sigurdur Helgason, Integral geometry and multitemporal wave equations, Comm. Pure Appl. Math. 51 (1998), no. 9-10, 1035–1071. Dedicated to the memory of Fritz John. MR 1632583, DOI 10.1002/(SICI)1097-0312(199809/10)51:9/10<1035::AID-CPA5>3.3.CO;2-H
  • I. M. Gel′fand, S. G. Gindikin, and M. I. Graev, Izbrannye zadachi integral′noĭ geometrii, Dobrosvet, Moscow, 2000 (Russian, with Russian summary). MR 1795833
  • Elena Ournycheva and Boris Rubin, An analogue of the Fuglede formula in integral geometry on matrix spaces, Complex analysis and dynamical systems II, Contemp. Math., vol. 382, Amer. Math. Soc., Providence, RI, 2005, pp. 305–320. MR 2175898, DOI 10.1090/conm/382/07070
  • I. G. Petrovskiĭ, Izbrannye trudy, “Nauka”, Moscow, 1986 (Russian). Sistemy uravneniĭ s chastnymi proizvodnymi. Algebraicheskaya geometriya. [Systems of partial differential equations. Algebraic geometry]; Edited and with a preface by V. I. Arnol′d, N. N. Bogolyubov, A. N. Kolmogorov, O. A. Oleĭnik, S. L. Sobolev and A. N. Tikhonov; Compiled by Oleĭnik; With commentaries by Kolmogorov, L. R. Volevich, V. Ya. Ivriĭ, I. M. Gel′fand, G. E. Shilov, Oleĭnik, V. P. Palamodov, A. M. Gabrièlov and V. M. Kharlamov. MR 871873
  • S. P. Khèkalo, Gauge-equivalent deformations of linear ordinary differential operators with constant coefficients, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 308 (2004), no. Mat. Vopr. Teor. Rasprostr. Voln. 33, 235–251, 256 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 132 (2006), no. 1, 136–145. MR 2092189, DOI 10.1007/s10958-005-0482-7
  • S. P. Khèkalo, The Cayley-Laplace differential operator on the space of rectangular matrices, Izv. Ross. Akad. Nauk Ser. Mat. 69 (2005), no. 1, 195–224 (Russian, with Russian summary); English transl., Izv. Math. 69 (2005), no. 1, 191–219. MR 2128187, DOI 10.1070/IM2005v069n01ABEH000528
  • N. Kh. Ibragimov and A. O. Oganesyan, Hierarchy of Huygens equations in spaces with a nontrivial conformal group, Uspekhi Mat. Nauk 46 (1991), no. 3(279), 111–146, 239 (Russian); English transl., Russian Math. Surveys 46 (1991), no. 3, 137–176. MR 1134091, DOI 10.1070/RM1991v046n03ABEH002795
  • S. P. Khèkalo, Stepwise gauge equivalence of differential operators, Mat. Zametki 77 (2005), no. 6, 917–929 (Russian, with Russian summary); English transl., Math. Notes 77 (2005), no. 5-6, 843–854. MR 2246966, DOI 10.1007/s11006-005-0084-1
  • Lars Gårding, The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals, Ann. of Math. (2) 48 (1947), 785–826. MR 22648, DOI 10.2307/1969381
  • S. P. Khèkalo, Homogeneous differential operators and Riesz potentials in the space of rectangular matrices, Dokl. Akad. Nauk 404 (2005), no. 5, 604–607 (Russian). MR 2256819
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Bibliographic Information
  • S. P. Khekalo
  • Affiliation: Kolomna State Pedagogical University, Russia
  • Email: fmf@kolomna.ru
  • Received by editor(s): September 21, 2007
  • Published electronically: August 22, 2008
  • Additional Notes: Supported by the president of RF (grant no. MK-2195.2007.1) and by RFBR (grant no. 07-01-00085).
  • © Copyright 2008 American Mathematical Society
  • Journal: St. Petersburg Math. J. 19 (2008), 1015-1028
  • MSC (2000): Primary 53A04; Secondary 52A40, 52A10
  • DOI: https://doi.org/10.1090/S1061-0022-08-01034-0
  • MathSciNet review: 2411965