|
Solution of the Hadamard problem in the class of stepwise gauge-equivalent deformations of homogeneous differential operators with constant coefficients
Author(s):
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
Posted:
August 22, 2008
MathSciNet review:
2411965
Retrieve article in:
PDF
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:
-
- 1.
- J. Hadamard, Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques, Hermann, Paris, 1932.
- 2.
- N. Kh. Ibragimov, Transformation groups in mathematical physics, ``Nauka'', Moscow, 1983. (Russian) MR 0734307 (85j:58003)
- 3.
- Yu. Yu. Berest and A. P. Veselov, The Huygens principle and integrability, Uspekhi Mat. Nauk 49 (1994), no. 6, 7-78; English transl., Russian Math. Surveys 49 (1994), no. 6, 5-77. MR 1316866 (96a:35003)
- 4.
- K. L. Stellmacher, Ein Beispiel einer Huyghensschen Differentialgleichung, Nachr. Akad. Wiss. Göttingen. Math. Phys. Kl. Math.-Phys. Chem. Abt. 1953, 133-138. MR 0060695 (15:710c)
- 5.
- J. E. Lagnese and K. L. Stellmacher, A method of generating classes of Huygens' operators, J. Math. Mech. 17 (1967), no. 5, 461-472. MR 0217409 (36:499)
- 6.
- G. Darboux, Sur la représentation sphérique des surfaces, Compt. Rend. (Paris) 94 (1882), 1343-1345.
- 7.
- Y. Berest and Y. Molchanov, Fundamental solutions for partial differential equations with reflection group invariance, J. Math. Phys. 36 (1995), 4324-4339. MR 1341994 (96c:35005)
- 8.
- Y. Berest, Hierarchies of Huygens' operators and Hadamard's conjecture, Acta Appl. Math. 53 (1998), no. 2, 125-185. MR 1646583 (99j:58204)
- 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; English transl., Izv. Math. 67 (2003), no. 4, 815-836. MR 2001768 (2004f:35109)
- 10.
- Y. Berest, The problem of lacunas and analysis on root systems, Trans. Amer. Math. Soc. 352 (2000), no. 8, 3743-3776. MR 1694280 (2001d:58030)
- 11.
- A. M. Gabrièlov and V. P. Palamodov, The Huygens' principle and its generalizations, I. G. Petrovskiĭ. Selected Works. Part I. Systems of Partial Differential Equations. Algebraic Geometry, ``Nauka'', Moscow, 1986, pp. 449-456; English transl., Classics of Soviet Mathematics, vol. 5, Gordon and Breach Publ., Amsterdam, 1996. MR 0871873 (88f:01059); MR 1677652 (99m:01106a)
- 12.
- S. G. Gindikin, The Cauchy problem for strongly homogeneous differential operators, Trudy Moskov. Mat. Obshch. 16 (1967), 181-208; English transl. in Trans. Moscow Math. Soc. 1967 (1968). MR 0227593 (37:3177)
- 13.
- V. M. Babich, Hadamard's ansatz, its analogues, generalizations and applications, Algebra i Analiz 3 (1991), no. 5, 1-37; English transl., St. Petersburg Math. J. 3 (1992), no. 5, 937-972. MR 1186234 (93i:35004)
- 14.
- P. Günther, Ein Beispiel einer nichttrivialen Huygensschen Differentialgleichung mit vier unabhängigen Variablen, Arch. Rational Mech. Anal. 18 (1965), 103-106. MR 0174865 (30:5056)
- 15.
- M. A. Semenov-Tyan-Shanskiĭ, Harmonic analysis on Riemannian symmetric spaces of negative curvature, and scattering theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 3, 562-592; English transl., Math. USSR-Izv. 10 (1976), no. 3, 535-563 (1977). MR 0467179 (57:7044)
- 16.
- S. Helgason, Integral geometry and multitemporal wave equations, Comm. Pure Appl. Math. 51 (1998), 1035-1071. MR 1632583 (99j:58207)
- 17.
- I. M. Gel'fand, S. G. Gindikin, and M. I. Graev, Selected problems in integral geometry, ``Dobrosvet'', Moscow, 2000. (Russian) MR 1795833 (2002d:53100)
- 18.
- E. Ournycheva and B. Rubin, An analogue of the Fuglede formula in integral geometry on matrix spaces, Preprint, Math.FA/0401127, 1, (2004), pp. 1-20. Contemp. Math., vol. 382, Amer. Math. Soc., Providence, RI, 2005, pp. 305-320. MR 2175898 (2006g:44005)
- 19.
- I. G. Petrovskiĭ, Selected works. Systems of partial differential equations. Algebraic geometry, ``Nauka'', Moscow, 1986; English transl., Gordon and Breach Publ., Amsterdam, 1996. MR 0871873 (88f:01059); MR 1677652 (99m:01106a)
- 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), 235-251; English transl., J. Math. Sci. (N. Y.) 132 (2006), no. 1, 136-145. MR 2092189 (2005g:34226)
- 21.
- -, The Cayley-Laplace differential operator on the space of rectangular matrices, Izv. Ross. Akad. Nauk Ser. Mat. 69 (2005), no. 1, 195-224; English transl., Izv. Math. 69 (2005), no. 1, 191-219. MR 2128187 (2005m:35009)
- 22.
- M. Kh. Ibragimov and A. O. Oganesyan, Hierarchy of Huygens equations in spaces with a nontrivial conformal group, Uspekhi Mat. Nauk 46 (1991), no. 3, 111-146; English transl., Russian Math. Surveys 46 (1991), no. 3, 137-176. MR 1134091 (92m:58137)
- 23.
- S. P. Khèkalo, Stepwise gauge equivalence of differential operators, Mat. Zametki 77 (2005), no. 6, 917-929; English transl., Math. Notes 77 (2005), no. 5-6, 843-854. MR 2246966 (2007d:35004)
- 24.
- L. 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 (9:240a)
- 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; English transl. in Russian Acad. Sci. Dokl. Math. MR 2256819 (2007h:22005)
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:
10.1090/S1061-0022-08-01034-0
PII:
S 1061-0022(08)01034-0
Keywords:
Hadamard problem,
Huygens principle,
homogeneous operators,
deformations,
Riesz kernels,
gauge equivalence,
stepwise gauge equivalence
Received by editor(s):
21/SEP/2007
Posted:
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 of article:
Copyright
2008,
American Mathematical Society
|
