Geodesic
Solution of the Hadamard problem in the class of stepwise gauge-equivalent deformations of homogeneous differential operators with constant coefficients
Algebra i analiz, Tome 19 (2007) no. 6, pp. 200-219.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: Hadamard problem, Huygens principle, homogeneous operators, deformations, Riesz kernels, gauge equivalence, stepwise gauge equivalence. 
@article{AA_2007_19_6_a9,
     author = {S. P. Khekalo},
     title = {Solution of the {Hadamard} problem in the class of stepwise gauge-equivalent deformations of homogeneous differential operators with constant coefficients},
     journal = {Algebra i analiz},
     pages = {200--219},
     publisher = {mathdoc},
     volume = {19},
     number = {6},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AA_2007_19_6_a9/}
}
TY  - JOUR
AU  - S. P. Khekalo
TI  - Solution of the Hadamard problem in the class of stepwise gauge-equivalent deformations of homogeneous differential operators with constant coefficients
JO  - Algebra i analiz
PY  - 2007
SP  - 200
EP  - 219
VL  - 19
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2007_19_6_a9/
LA  - ru
ID  - AA_2007_19_6_a9
ER  - 
%0 Journal Article
%A S. P. Khekalo
%T Solution of the Hadamard problem in the class of stepwise gauge-equivalent deformations of homogeneous differential operators with constant coefficients
%J Algebra i analiz
%D 2007
%P 200-219
%V 19
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2007_19_6_a9/
%G ru
%F AA_2007_19_6_a9
S. P. Khekalo. Solution of the Hadamard problem in the class of stepwise gauge-equivalent deformations of homogeneous differential operators with constant coefficients. Algebra i analiz, Tome 19 (2007) no. 6, pp. 200-219. http://geodesic.mathdoc.fr/item/AA_2007_19_6_a9/

[1] Adamar Zh., Zadacha Koshi dlya lineinykh uravnenii s chastnymi proizvodnymi giperbolicheskogo tipa, Nauka, M., 1978 | MR

[2] Ibragimov N. Kh., Gruppy preobrazovanii v matematicheskoi fizike, Nauka, M., 1983 | MR

[3] Berest Yu. Yu., Veselov A. P., “Printsip Gyuigensa i integriruemost”, UMN, 49:6 (1994), 7–78 | MR | Zbl

[4] Stellmacher K. L., “Ein Beispiel einer Huyghensschen Differentialgleichung”, Nachr. Akad. Wiss. Göttingen. Math. Phys. Kl. Math.-Phys. Chem. Abt., 1953 (1953), 133–138 | MR | Zbl

[5] Lagnese J. E., Stellmacher K. L., “A method of generating classes of Huygens' operators”, J. Math. Mech., 17:5 (1967), 461–472 | MR | Zbl

[6] Darboux G., “Sur la representations spherique des surfaces”, Compt. Rend. (Paris), 94 (1882), 1343–1345

[7] Berest Y., Molchanov Y., “Fundamental solutions for partial differential equations with reflection group invariance”, J. Math. Phys., 36 (1995), 4324–4339 | DOI | MR | Zbl

[8] Berest Y., “Hierarchies of Huygens' operators and Hadamard's conjecture”, Acta Appl. Math., 53:2 (1998), 125–185 | DOI | MR | Zbl

[9] Khekalo S. P., “Izogyuigensovy deformatsii operatora Keli obschim potentsialom Lagneze–Shtelmakhera”, Izv. RAN. Ser. mat., 67:4 (2003), 189–212 | MR | Zbl

[10] Berest Y., “The problem of lacunas and analysis on root systems”, Trans. Amer. Math. Soc., 352:8 (2000), 3743–3776 | DOI | MR | Zbl

[11] Gabrielov A. M., Palamodov V. P., “Printsip Gyuigensa i ego obobscheniya”, Izbrannye trudy. Sistemy uravnenii s chastnymi proizvodnymi. Algebraicheskaya geometriya, ed. I. G. Petrovskii, Nauka, M., 1986, 449–456 | MR

[12] Gindikin S. G., “Zadacha Koshi dlya silno odnorodnykh differentsialnykh operatorov”, Tr. Mosk. mat. o-va, 16, 1967, 181–208 | MR | Zbl

[13] Babich V. M., “Anzatts Adamara, ego analogi, obobscheniya, prilozheniya”, Algebra i analiz, 3:5 (1991), 1–37 | MR

[14] Günther P., “Ein Beispiel einer nichttrivialen Huygensschen Differentialgleichung mit vier unabhängigen Variablen”, Arch. Rational Mech. Anal., 18 (1965), 103–106 | MR | Zbl

[15] Semenov-Tyan-Shanskii M. A., “Garmonicheskii analiz na rimanovykh simmetricheskikh prostranstvakh otritsatelnoi krivizny i teoriya rasseyaniya”, Izv. AN SSSR. Ser. mat., 40:3 (1976), 562–592 | MR

[16] Helgason S., “Integral geometry and multitemporal wave equation”, Comm. Pure Appl. Math., 51 (1998), 1035–1071 | 3.0.CO;2-E class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[17] Gelfand I. M., Gindikin S. G., Graev M. I., Izbrannye zadachi integralnoi geometrii, Dobrosvet, M., 2000 | MR

[18] Ournycheva E., Rubin B., An analogue of the Fuglede formula in integral geometry on matrix space, Preprint ; , 2004 arXiv: /math/0401127 | MR

[19] Petrovskii I. G., Izbrannye trudy. Sistemy uravnenii s chastnymi proizvodnymi. Algebraicheskaya geometriya, Nauka, M., 1986 | MR | Zbl

[20] Khekalo S. P., “Kalibrovochno ekvivalentnye deformatsii obyknovennykh lineinykh differentsialnykh operatorov s postoyannymi koeffitsientami”, Zap. nauchn. sem. POMI, 308, 2004, 235–251 | Zbl

[21] Khekalo S. P., “Differentsialnyi operator Keli–Laplasa na prostranstve pryamougolnykh matrits”, Izv. RAN. Ser. mat., 69:1 (2005), 195–224 | MR | Zbl

[22] Ibragimov N. Kh., Oganesyan A. O., “Ierarkhiya gyuigensovykh uravnenii v prostranstvakh s netrivialnoi konformnoi gruppoi”, UMN, 46:3(279) (1991), 111–146 | MR | Zbl

[23] Khekalo S. P., “Poshagovaya kalibrovochnaya ekvivalentnost differentsialnykh operatorov”, Matem. zametki, 77:6 (2005), 917–929 | MR | Zbl

[24] Gårding L., “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 | DOI | MR

[25] Khekalo S. P., “Odnorodnye differentsialnye operatory i potentsialy Rissa na prostranstve pryamougolnykh matrits”, Dokl. RAN, 404:5 (2005), 604–607 | MR | Zbl