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.
Biography
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:
R.R. Gadylshin, “Suschestvovanie i asimptotiki polyusov s maloi mnimoi chastyu dlya rezonatora Gelmgoltsa”, Uspekhi matematicheskikh nauk, 52:1(313) (1997), 3–76
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; Izv. Math., 82:6 (2018), 1108–1135
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; Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 76–90
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; Ufa Math. J., 7:3 (2015), 15–27
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; Proc. Steklov Inst. Math. (Suppl.), 281, suppl. 1 (2013), 44–58
R. R. Gadyl'shin, I. Kh. Khusnullin, “Perturbation of a periodic operator by a narrow potential”, TMF, 173:1 (2012), 127–134; Theoret. and Math. Phys., 173:1 (2012), 1438–1444
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
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, no. 14, 27–36
2010
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; St. Petersburg Math. J., 22:6 (2011), 883–894
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; Izv. Math., 72:4 (2008), 659–688
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; Sb. Math., 198:8 (2007), 1063–1093
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; Math. Notes, 79:5 (2006), 729–733
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; Theoret. and Math. Phys., 147:1 (2006), 496–500
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; Comput. Math. Math. Phys., 46:1 (2006), 97–110
R. R. Gadyl'shin, “Local Perturbations of the Schrödinger Operator on the Plane”, TMF, 138:1 (2004), 41–54; Theoret. and Math. Phys., 138:1 (2004), 33–44
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; Proc. Steklov Inst. Math. (Suppl.), 2003no. , suppl. 1, S68–S74
R. R. Gadyl'shin, “On a Model Analogue of the Helmholtz Resonator in Homogenization”, Trudy Mat. Inst. Steklova, 236 (2002), 79–86; Proc. Steklov Inst. Math., 236 (2002), 70–77
R. R. Gadyl'shin, “Local Perturbations of the Schrödinger Operator on the Axis”, TMF, 132:1 (2002), 97–104; Theoret. and Math. Phys., 132:1 (2002), 976–982
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; Comput. Math. Math. Phys., 41:12 (2001), 1765–1776
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; Sb. Math., 191:6 (2000), 821–848
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; Differ. Equ., 35:4 (1999), 540–551
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; Siberian Math. J., 40:2 (1999), 229–244
R. R. Gadyl'shin, O. M. Kiselev, “Structural instability of a soliton for the Davey–Stewartson II equation”, TMF, 118:3 (1999), 354–361; Theoret. and Math. Phys., 118:3 (1999), 278–284
D. I. Borisov, R. R. Gadyl'shin, “On the spectrum of the Laplacian with frequently alternating boundary conditions”, TMF, 118:3 (1999), 347–353; Theoret. and Math. Phys., 118:3 (1999), 272–277
R. R. Gadyl'shin, “On the asymptotics of eigenvalues for a periodically fixed membrane”, Algebra i Analiz, 10:1 (1998), 3–19; St. Petersburg Math. J., 10:1 (1999), 1–14
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; Sb. Math., 189:4 (1998), 503–526
R. R. Gadyl'shin, “Characteristic frequencies of bodies with thin spikes. III. Frequency splitting”, Mat. Zametki, 61:4 (1997), 494–502; Math. Notes, 61:4 (1997), 409–416
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; Russian Math. Surveys, 52:1 (1997), 1–72
R. R. Gadyl'shin, “On scattering by a cylindrical surface with a narrow crack”, Differ. Uravn., 32:10 (1996), 1362–1370; 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; Theoret. and Math. Phys., 106:2 (1996), 167–173
R. R. Gadyl'shin, “On scattering by cylinder with narrow slit and with shell of finite depth”, TMF, 106:1 (1996), 24–43; Theoret. and Math. Phys., 106:1 (1996), 19–34
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; Comput. Math. Math. Phys., 36:7 (1996), 889–898
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; Comput. Math. Math. Phys., 36:1 (1996), 75–84
R. R. Gadyl'shin, “A Complete Electromagnetic Analog of the Helmholtz Resonator”, Funktsional. Anal. i Prilozhen., 29:4 (1995), 1–16; Funct. Anal. Appl., 29:4 (1995), 225–237
1994
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; 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; Differ. Equ., 30:2 (1994), 201–209
R. R. Gadyl'shin, “On eigenfrequencies of bodies with thin branches. II. Asymptotics”, Mat. Zametki, 55:1 (1994), 20–34; Math. Notes, 55:1 (1994), 14–23
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; J. Math. Sci., 83:2 (1997), 204–209
1993
46.
R. R. Gadyl'shin, “On the Poles of an Acoustic Resonator”, Funktsional. Anal. i Prilozhen., 27:4 (1993), 3–16; Funct. Anal. Appl., 27:4 (1993), 229–239
R. R. Gadyl'shin, “Splitting of the poles of a Helmholtz resonator”, Izv. RAN. Ser. Mat., 57:5 (1993), 44–74; Russian Acad. Sci. Izv. Math., 43:2 (1994), 233–260
R. R. Gadyl'shin, “Characteristic frequencies of bodies with thin spikes. I. Convergence and estimates”, Mat. Zametki, 54:6 (1993), 10–21; Math. Notes, 54:6 (1993), 1192–1199
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; Siberian Math. J., 34:3 (1993), 433–450
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; Theoret. and Math. Phys., 97:1 (1993), 1154–1159
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; St. Petersburg Math. J., 4:2 (1993), 273–296
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; Math. Notes, 52:4 (1992), 1020–1029
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; Theoret. and Math. Phys., 93:1 (1992), 1151–1159
R. R. Gadyl'shin, “The Helmholtz resonator with a small aperture”, Zh. Vychisl. Mat. Mat. Fiz., 32:9 (1992), 1464–1475; Comput. Math. Math. Phys., 32:9 (1992), 1311–1321
R. R. Gadyl'shin, “The amplitude of oscillations for a Helmholtz resonator”, Dokl. Akad. Nauk SSSR, 310:5 (1990), 1094–1097; Dokl. Math., 35:2 (1990), 141–143
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
D. I. Borisov, R. R. Gadyl'shin, “Errata”, Mat. Sb., 199:3 (2008), 160
2002
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; Differ. Equ., 38:8 (2002), 1075–1080