Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

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

 

 

Solution of the Hadamard problem in the class of stepwise gauge-equivalent deformations of homogeneous differential operators with constant coefficients


Author: S. P. Khekalo
Translated by: the author
Original publication: Algebra i Analiz, tom 19 (2007), nomer 6.
Journal: St. Petersburg Math. J. 19 (2008), 1015-1028
MSC (2000): Primary 53A04; Secondary 52A40, 52A10
Published electronically: August 22, 2008
MathSciNet review: 2411965
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. J. Hadamard, Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques, Hermann, Paris, 1932.
  • 2. N. Kh. Ibragimov, Gruppy preobrazovanii v matematicheskoi fizike, “Nauka”, Moscow, 1983 (Russian). MR 734307
  • 3. 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, 10.1070/RM1994v049n06ABEH002447
  • 4. 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 0060695
  • 5. J. E. Lagnese and K. L. Stellmacher, A method of generating classes of Huygens’ operators, J. Math. Mech. 17 (1967), 461–472. MR 0217409
  • 6. G. Darboux, Sur la représentation sphérique des surfaces, Compt. Rend. (Paris) 94 (1882), 1343-1345.
  • 7. 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, 10.1063/1.530964
  • 8. Yuri Berest, Hierarchies of Huygens’ operators and Hadamard’s conjecture, Acta Appl. Math. 53 (1998), no. 2, 125–185. MR 1646583, 10.1023/A:1006069012474
  • 9. 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, 10.1070/IM2003v067n04ABEH000446
  • 10. Yuri Berest, The problem of lacunas and analysis on root systems, Trans. Amer. Math. Soc. 352 (2000), no. 8, 3743–3776. MR 1694280, 10.1090/S0002-9947-00-02543-5
  • 11. I. G. Petrovskiĭ, Izbrannye trudy, “Nauka”, Moscow, 1986 (Russian). Sistemy uravnenii 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
    I. G. Petrowsky, Selected works. Part I, Classics of Soviet Mathematics, vol. 5, Gordon and Breach Publishers, Amsterdam, 1996. Systems of partial differential equations and algebraic geometry; Introductory material by A. N. Kolmogorov and O. A. Oleinik; Translated from the Russian by G. A. Yosifian [G. A. Iosif′yan]; With a foreword by Lars Gårding; Edited and with a preface by Oleinik. MR 1677652
  • 12. S. G. Gindikin, The Cauchy problem for strongly homogeneous differential operators, Trudy Moskov. Mat. Obšč. 16 (1967), 181–208 (Russian). MR 0227593
  • 13. 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
  • 14. Paul Günther, Ein Beispiel einer nichttrivialen Huygensschen Differentialgleichung mit vier unabhängigen Variablen, Arch. Rational Mech. Anal. 18 (1965), 103–106 (German). MR 0174865
  • 15. 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
  • 16. 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, 10.1002/(SICI)1097-0312(199809/10)51:9/10<1035::AID-CPA5>3.3.CO;2-H
  • 17. I. M. Gel′fand, S. G. Gindikin, and M. I. Graev, Izbrannye zadachi integralnoi geometrii, Dobrosvet, Moscow, 2000 (Russian, with Russian summary). MR 1795833
  • 18. 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, 10.1090/conm/382/07070
  • 19. I. G. Petrovskiĭ, Izbrannye trudy, “Nauka”, Moscow, 1986 (Russian). Sistemy uravnenii 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
    I. G. Petrowsky, Selected works. Part I, Classics of Soviet Mathematics, vol. 5, Gordon and Breach Publishers, Amsterdam, 1996. Systems of partial differential equations and algebraic geometry; Introductory material by A. N. Kolmogorov and O. A. Oleinik; Translated from the Russian by G. A. Yosifian [G. A. Iosif′yan]; With a foreword by Lars Gårding; Edited and with a preface by Oleinik. MR 1677652
  • 20. 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, 10.1007/s10958-005-0482-7
  • 21. 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, 10.1070/IM2005v069n01ABEH000528
  • 22. 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, 10.1070/RM1991v046n03ABEH002795
  • 23. 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, 10.1007/s11006-005-0084-1
  • 24. 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 0022648
  • 25. 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

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 53A04, 52A40, 52A10

Retrieve articles in all journals with MSC (2000): 53A04, 52A40, 52A10


Additional Information

S. P. Khekalo
Affiliation: Kolomna State Pedagogical University, Russia
Email: fmf@kolomna.ru

DOI: https://doi.org/10.1090/S1061-0022-08-01034-0
Keywords: Hadamard problem, Huygens principle, homogeneous operators, deformations, Riesz kernels, gauge equivalence, stepwise gauge equivalence
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).
Article copyright: © Copyright 2008 American Mathematical Society