Gadyl'shin, Rustem Rashitovich

Statistics Math-Net.Ru
Total publications: 60
Scientific articles: 58

Number of views:
This page:2065
Abstract pages:15346
Full texts:5675
Doctor of physico-mathematical sciences (1995)
Speciality: 01.01.02 (Differential equations, dynamical systems, and optimal control)
Birth date: 14.04.1955
Keywords: singular perturbations, asymptotics, boundary value problems, Laplacian, eigenvalues, scattering problems, Helmholtz resonator, homoggenization.
UDC: 517.956, 517.958, 531.33, 535.4, 517.95, 517.9, 517.98, 517.984, 519.632.4, 517.928
MSC: 35J05, 35J25, 35p25, 47A45, 35P15, 58J37, 58J50, 35C20, 35J10, 35B99, 35P20, 47A40, 76Q05, 70F05


Asymptotics of eigenvalues with respect of a small parameter of singular perturbed boundary value problems for Laplacian in domains with narrow appendixes, connecting channels, slits, with exchanging of a type boundary conditions at small parts of boundaries and with frequently alternating changing of the boundary conditions are constructed. For Helmholtz resonators of the various types, the existence of poles with small imaginary parts for analytic continuations of solutions is proved. Asymtotics of these pole are constructed and treir resonant nature is shown. The existence of an analog of Helmholtz resonator in the hogomonization theo.


Graduated from Faculty of Mathematics of Bashkir State University in 1977 (department of theory of differential equation). Ph.D. thesis was defended in 1988. D.Sci. thesis was defended in 1995. A list of my works contains more than 90 titles.

Main publications:
  1. R.R. Gadylshin, “Suschestvovanie i asimptotiki polyusov s maloi mnimoi chastyu dlya rezonatora Gelmgoltsa”, Uspekhi matematicheskikh nauk, 52:1(313) (1997), 3–76  mathnet  mathscinet  zmath
  2. R.R. Gadylshin, “Ob asimptotike sobstvennykh znachenii dlya periodicheski zakreplennoi membrany”, Algebra i analiz, 10:1 (1998), 3–19  mathnet  mathscinet  zmath
  3. R.R. Gadylshin, A.M. Ilin, “Asimptotika sobstvennykh znachenii zadachi Dirikhle v oblasti s uzkoi schelyu”, Matematicheskii sbornik, 189:4 (1998), 25–48  mathnet  mathscinet  zmath
  4. A.G. Belyaev, G.A. Chechkin, R.R. Gadyl'shin, “Effective membrane permeability: Estimates and low concentration asymptotics”, SIAM J. Appl. Math., 60:1 (2000), 84–108  crossref  mathscinet  zmath
  5. R.R. Gadylshin, “Sistemy rezonatorov”, Izv. RAN, ser. matem., 64:3 (2000), 51–96  mathnet  mathscinet  zmath
List of publications on Google Scholar
List of publications on ZentralBlatt

Publications in Math-Net.Ru
1. R. R. Gadyl'shin, A. L. Piatnitski, G. A. Chechkin, “On the asymptotic behaviour of eigenvalues of a boundary-value problem in a planar domain of Steklov sieve type”, Izv. RAN. Ser. Mat., 82:6 (2018),  37–64  mathnet  elib; Izv. Math., 82:6 (2018), 1108–1135  isi  scopus
2. R. R. Gadyl'shin, S. V. Repjevskij, E. A. Shishkina, “On an eigenvalue for the Laplace operator in a disk with Dirichlet boundary condition on a small part of the boundary in a critical case”, Trudy Inst. Mat. i Mekh. UrO RAN, 21:1 (2015),  56–70  mathnet  mathscinet  elib; Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 76–90  isi  scopus
3. R. R. Gadylshin, A. A. Ershov, S. V. Repyevsky, “On asymptotic formula for electric resistance of conductor with small contacts”, Ufimsk. Mat. Zh., 7:3 (2015),  16–28  mathnet  elib; Ufa Math. J., 7:3 (2015), 15–27  isi  scopus
4. R. R. Gadyl'shin, E. A. Shishkina, “On Friedrichs inequalities for a disk”, Trudy Inst. Mat. i Mekh. UrO RAN, 18:2 (2012),  48–61  mathnet  elib; Proc. Steklov Inst. Math. (Suppl.), 281, suppl. 1 (2013), 44–58  isi  scopus
5. R. R. Gadyl'shin, I. Kh. Khusnullin, “Perturbation of a periodic operator by a narrow potential”, TMF, 173:1 (2012),  127–134  mathnet  mathscinet  elib; Theoret. and Math. Phys., 173:1 (2012), 1438–1444  isi  elib  scopus
6. A. R. Bikmetov, R. R. Gadyl'shin, “Perturbation of an elliptic operator by a narrow potential in an $n$-dimensional domain”, Ufimsk. Mat. Zh., 4:2 (2012),  28–64  mathnet  mathscinet
7. R. R. Gadyl'shin, I. Kh. Khusnullin, “Perturbation of the Shrödinger operator by a narrow potential”, Ufimsk. Mat. Zh., 3:3 (2011),  55–66  mathnet  zmath
8. R. R. Gadyl'shin, Yu. O. Koroleva, G. A. Chechkin, “On the asymptotics of a solution of a boundary value problem in a domain perforated along boundary”, Vestnik Chelyabinsk. Gos. Univ., 2011, 14,  27–36  mathnet
9. R. R. Gadyl'shin, I. Kh. Khusnullin, “Schrödinger operator on the axis with potentials depending on two parameters”, Algebra i Analiz, 22:6 (2010),  50–66  mathnet  mathscinet  zmath; St. Petersburg Math. J., 22:6 (2011), 883–894  isi  scopus
10. D. I. Borisov, R. R. Gadyl'shin, “On the spectrum of a periodic operator with a small localized perturbation”, Izv. RAN. Ser. Mat., 72:4 (2008),  37–66  mathnet  mathscinet  zmath  elib; Izv. Math., 72:4 (2008), 659–688  isi  elib  scopus
11. D. I. Borisov, R. R. Gadyl'shin, “The spectrum of a self-adjoint differential operator with rapidly oscillating coefficients on the axis”, Mat. Sb., 198:8 (2007),  3–34  mathnet  mathscinet  zmath  elib; Sb. Math., 198:8 (2007), 1063–1093  isi  elib  scopus
12. A. R. Bikmetov, R. R. Gadyl'shin, “On the spectrum of the Schrödinger operator with large potential concentrated on a small set”, Mat. Zametki, 79:5 (2006),  787–790  mathnet  mathscinet  zmath  elib; Math. Notes, 79:5 (2006), 729–733  isi  scopus
13. D. I. Borisov, R. R. Gadyl'shin, “The spectrum of the Schrödinger operator with a rapidly oscillating compactly supported potential”, TMF, 147:1 (2006),  58–63  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 147:1 (2006), 496–500  isi  scopus
14. Y. Amirat, G. A. Chechkin, R. R. Gadyl'shin, “Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with oscillating boundary”, Zh. Vychisl. Mat. Mat. Fiz., 46:1 (2006),  102–115  mathnet  mathscinet  zmath; Comput. Math. Math. Phys., 46:1 (2006), 97–110  scopus
15. R. R. Gadyl'shin, “On the eigenvalues of a “dumb-bell with a thin handle””, Izv. RAN. Ser. Mat., 69:2 (2005),  45–110  mathnet  mathscinet  zmath  elib; Izv. Math., 69:2 (2005), 265–329  isi  elib  scopus
16. R. R. Gadyl'shin, “Local Perturbations of Quantum Waveguides”, TMF, 145:3 (2005),  358–371  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 145:3 (2005), 1678–1690  isi
17. R. R. Gadyl'shin, “Local Perturbations of the Schrödinger Operator on the Plane”, TMF, 138:1 (2004),  41–54  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 138:1 (2004), 33–44  isi
18. R. R. Gadyl'shin, “Scattering by a Cylindrical Trap in the Critical Case”, Mat. Zametki, 73:3 (2003),  355–370  mathnet  mathscinet  zmath; Math. Notes, 73:3 (2003), 328–341  isi  scopus
19. R. R. Gadyl'shin, “On domains with a narrow isthmus in the critical case”, Trudy Inst. Mat. i Mekh. UrO RAN, 9:1 (2003),  50–55  mathnet  mathscinet  zmath  elib; Proc. Steklov Inst. Math. (Suppl.), 2003no. , suppl. 1, S68–S74
20. R. R. Gadyl'shin, O. M. Kiselev, “Perturbation of soliton for Davey–Stewartson II equation”, Mat. Fiz. Anal. Geom., 9:2 (2002),  272–280  mathnet  mathscinet  zmath
21. R. R. Gadyl'shin, “Analogues of the Helmholtz resonator in homogenization theory”, Mat. Sb., 193:11 (2002),  43–70  mathnet  mathscinet  zmath  elib; Sb. Math., 193:11 (2002), 1611–1638  isi  scopus
22. R. R. Gadyl'shin, “On a Model Analogue of the Helmholtz Resonator in Homogenization”, Tr. Mat. Inst. Steklova, 236 (2002),  79–86  mathnet  mathscinet  zmath; Proc. Steklov Inst. Math., 236 (2002), 70–77
23. R. R. Gadyl'shin, “Local Perturbations of the Schrödinger Operator on the Axis”, TMF, 132:1 (2002),  97–104  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 132:1 (2002), 976–982  isi
24. R. R. Gadyl'shin, “Homogenization and asymptotics for a membrane with closely spaced clamping points”, Zh. Vychisl. Mat. Mat. Fiz., 41:12 (2001),  1857–1869  mathnet  mathscinet  zmath  elib; Comput. Math. Math. Phys., 41:12 (2001), 1765–1776
25. R. R. Gadyl'shin, “Resonator systems”, Izv. RAN. Ser. Mat., 64:3 (2000),  51–96  mathnet  mathscinet  zmath; Izv. Math., 64:3 (2000), 487–529  isi  scopus
26. R. R. Gadyl'shin, “On the Dirichlet problem for the Helmholtz equation on the plane with boundary conditions on an almost closed curve”, Mat. Sb., 191:6 (2000),  43–68  mathnet  mathscinet  zmath; Sb. Math., 191:6 (2000), 821–848  isi  scopus
27. R. R. Gadyl'shin, “Asymptotics of the eigenvalues of a boundary value problem with rapidly oscillating boundary conditions”, Differ. Uravn., 35:4 (1999),  540–551  mathnet  mathscinet; Differ. Equ., 35:4 (1999), 540–551
28. R. R. Gadyl'shin, G. A. Chechkin, “A boundary value problem for the Laplacian with rapidly changing type of boundary conditions in a multi-dimensional domain”, Sibirsk. Mat. Zh., 40:2 (1999),  271–287  mathnet  mathscinet  zmath; Siberian Math. J., 40:2 (1999), 229–244  isi
29. R. R. Gadyl'shin, O. M. Kiselev, “Structural instability of a soliton for the Davey–Stewartson II equation”, TMF, 118:3 (1999),  354–361  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 118:3 (1999), 278–284  isi
30. D. I. Borisov, R. R. Gadyl'shin, “On the spectrum of the Laplacian with frequently alternating boundary conditions”, TMF, 118:3 (1999),  347–353  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 118:3 (1999), 272–277  isi
31. R. R. Gadyl'shin, “On the asymptotics of eigenvalues for a periodically fixed membrane”, Algebra i Analiz, 10:1 (1998),  3–19  mathnet  mathscinet  zmath; St. Petersburg Math. J., 10:1 (1999), 1–14
32. R. R. Gadyl'shin, A. M. Il'in, “Asymptotic behaviour of the eigenvalues of the Dirichlet problem in a domain with a narrow slit”, Mat. Sb., 189:4 (1998),  25–48  mathnet  mathscinet  zmath  elib; Sb. Math., 189:4 (1998), 503–526  isi  scopus
33. R. R. Gadyl'shin, “Characteristic frequencies of bodies with thin spikes. III. Frequency splitting”, Mat. Zametki, 61:4 (1997),  494–502  mathnet  mathscinet  zmath  elib; Math. Notes, 61:4 (1997), 409–416  isi
34. R. R. Gadyl'shin, “Existence and asymptotics of poles with small imaginary part for the Helmholtz resonator”, Uspekhi Mat. Nauk, 52:1(313) (1997),  3–76  mathnet  mathscinet  zmath; Russian Math. Surveys, 52:1 (1997), 1–72  isi  scopus
35. R. R. Gadyl'shin, “On scattering by a cylindrical surface with a narrow crack”, Differ. Uravn., 32:10 (1996),  1362–1370  mathnet  mathscinet; Differ. Equ., 32:10 (1996), 1358–1367
36. R. R. Gadyl'shin, O. M. Kiselev, “On nonsolution structure of scattering data under perturbation of two-dimensional soliton for Davey–Stewartson equation II”, TMF, 106:2 (1996),  200–208  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 106:2 (1996), 167–173  isi
37. R. R. Gadyl'shin, “On scattering by cylinder with narrow slit and with shell of finite depth”, TMF, 106:1 (1996),  24–43  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 106:1 (1996), 19–34  isi
38. R. R. Gadyl'shin, “On the perturbation of the Laplacian spectrum when the boundary condition type changes on a small part of the boundary”, Zh. Vychisl. Mat. Mat. Fiz., 36:7 (1996),  77–88  mathnet  mathscinet  zmath; Comput. Math. Math. Phys., 36:7 (1996), 889–898  isi
39. R. R. Gadyl'shin, “Asymptotic expansion of the solutions of a singularly perturbed Dirichlet problem”, Zh. Vychisl. Mat. Mat. Fiz., 36:1 (1996),  92–102  mathnet  mathscinet  zmath; Comput. Math. Math. Phys., 36:1 (1996), 75–84  isi
40. R. R. Gadyl'shin, “A Complete Electromagnetic Analog of the Helmholtz Resonator”, Funktsional. Anal. i Prilozhen., 29:4 (1995),  1–16  mathnet  mathscinet  zmath; Funct. Anal. Appl., 29:4 (1995), 225–237  isi
41. R. R. Gadyl'shin, “On the existence of poles with a small imaginary part for a complete electromagnetic analogue of the Helmholtz resonator”, Dokl. Akad. Nauk, 338:2 (1994),  151–153  mathnet  mathscinet  zmath; Dokl. Math., 50:2 (1995), 200–203
42. R. R. Gadyl'shin, “A two-dimensional analogue of the Helmholtz resonator with ideally rigid walls”, Differ. Uravn., 30:2 (1994),  221–229  mathnet  mathscinet; Differ. Equ., 30:2 (1994), 201–209
43. R. R. Gadyl'shin, “On eigenfrequencies of bodies with thin branches. II. Asymptotics”, Mat. Zametki, 55:1 (1994),  20–34  mathnet  mathscinet  zmath  elib; Math. Notes, 55:1 (1994), 14–23  isi
44. R. R. Gadyl'shin, “On scattering by bodies with narrow channels”, Mat. Sb., 185:7 (1994),  39–62  mathnet  mathscinet  zmath; Russian Acad. Sci. Sb. Math., 82:2 (1995), 293–313  isi
45. R. R. Gadyl'shin, “On scattering of $H$-polarized electromagnetic field by ideal conductive cylinder with narrow lengthwise split of finite depth”, Zap. Nauchn. Sem. POMI, 210 (1994),  65–72  mathnet  mathscinet  zmath; J. Math. Sci., 83:2 (1997), 204–209
46. R. R. Gadyl'shin, “On the Poles of an Acoustic Resonator”, Funktsional. Anal. i Prilozhen., 27:4 (1993),  3–16  mathnet  mathscinet  zmath; Funct. Anal. Appl., 27:4 (1993), 229–239  isi
47. R. R. Gadyl'shin, “Splitting of the poles of a Helmholtz resonator”, Izv. RAN. Ser. Mat., 57:5 (1993),  44–74  mathnet  mathscinet  zmath; Russian Acad. Sci. Izv. Math., 43:2 (1994), 233–260  isi
48. R. R. Gadyl'shin, “Characteristic frequencies of bodies with thin spikes. I. Convergence and estimates”, Mat. Zametki, 54:6 (1993),  10–21  mathnet  mathscinet  zmath; Math. Notes, 54:6 (1993), 1192–1199  isi
49. R. R. Gadyl'shin, “Splitting a multiple eigenvalue in the boundary value problem for a membrane clamped on a small part of the boundary”, Sibirsk. Mat. Zh., 34:3 (1993),  43–61  mathnet  mathscinet  zmath; Siberian Math. J., 34:3 (1993), 433–450  isi
50. R. R. Gadyl'shin, “Asymptotic behavior of the second eigenfrequency for a system of two bodies connected by a thin rod”, TMF, 97:1 (1993),  68–77  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 97:1 (1993), 1154–1159  isi
51. R. R. Gadyl'shin, “Surface potentials and the method of matching asymptotic expansions in the Helmholtz resonator problem”, Algebra i Analiz, 4:2 (1992),  88–115  mathnet  mathscinet  zmath; St. Petersburg Math. J., 4:2 (1993), 273–296
52. R. R. Gadyl'shin, “On a system of embedded resonators”, Dokl. Akad. Nauk, 326:6 (1992),  939–942  mathnet  mathscinet  zmath; Dokl. Math., 46:2 (1993), 383–386
53. R. R. Gadyl'shin, “On merging poles of an acoustic resonator”, Dokl. Akad. Nauk, 324:4 (1992),  773–776  mathnet  mathscinet; Dokl. Math., 37:6 (1992), 281–282
54. R. R. Gadyl'shin, “Ramification of a multiple eigenvalue of the Dirichlet problem for the Laplacian under singular perturbation of the boundary condition”, Mat. Zametki, 52:4 (1992),  42–55  mathnet  mathscinet  zmath; Math. Notes, 52:4 (1992), 1020–1029  isi
55. R. R. Gadyl'shin, “Influence of the position of the opening and its shape on the properties of a Helmholtz resonator”, TMF, 93:1 (1992),  107–118  mathnet  mathscinet; Theoret. and Math. Phys., 93:1 (1992), 1151–1159  isi
56. R. R. Gadyl'shin, “The Helmholtz resonator with a small aperture”, Zh. Vychisl. Mat. Mat. Fiz., 32:9 (1992),  1464–1475  mathnet  mathscinet  zmath; Comput. Math. Math. Phys., 32:9 (1992), 1311–1321  isi
57. R. R. Gadyl'shin, “The amplitude of oscillations for a Helmholtz resonator”, Dokl. Akad. Nauk SSSR, 310:5 (1990),  1094–1097  mathnet  mathscinet; Dokl. Math., 35:2 (1990), 141–143
58. R. R. Gadyl'shin, “Asymptotics of the eigenvalue of a singularly perturbed selfadjoint elliptic problem with a small parameter in the boundary conditions”, Differ. Uravn., 22:4 (1986),  640–652  mathnet  mathscinet

59. D. I. Borisov, R. R. Gadyl'shin, “Errata”, Mat. Sb., 199:3 (2008),  160  mathnet  mathscinet  elib
60. V. M. Babich, R. R. Gadyl'shin, A. R. Danilin, S. Yu. Dobrokhotov, V. A. Il'in, L. A. Kalyakin, E. F. Mishchenko, V. Yu. Novokshenov, Yu. S. Osipov, M. D. Ramazanov, N. Kh. Rozov, V. A. Sadovnichii, “Arlen Mikhailovich Il'in (A tribute in honor of his 70th birthday)”, Differ. Uravn., 38:8 (2002),  1011–1016  mathnet  mathscinet; Differ. Equ., 38:8 (2002), 1075–1080

Contact us:
 Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020