메뉴 건너뛰기




Volumn 42, Issue 6, 2010, Pages 2581-2609

The localization effect for eigenfunctions of the mixed boundary value problem in a thin cylinder with distorted ends

Author keywords

Boundary layer; Localization of eigenfunctions; Spectral problem; Thin domain; Trapped modes

Indexed keywords

DIRICHLET; DIRICHLET CONDITION; EIGEN FUNCTION; LAPLACE OPERATOR; LOCALIZATION EFFECT; MIXED BOUNDARY-VALUE PROBLEM; MIXED-BOUNDARY; NEUMANN PROBLEM; SPECTRAL PROBLEM; SUFFICIENT CONDITIONS; THIN DOMAINS; TRAPPED MODES;

EID: 79251497326     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/090755680     Document Type: Article
Times cited : (34)

References (68)
  • 2
    • 0003521938 scopus 로고
    • Springer-Verlag, New York, in English
    • Applied Mathematical Sciences 49, Springer-Verlag, New York, 1985 (in English).
    • (1985) Applied Mathematical Sciences 49
  • 4
    • 35648955038 scopus 로고    scopus 로고
    • Embedded trapped modes in water waves and acustics
    • C.M. LINTON AND P. MCIVER, Embedded trapped modes in water waves and acustics, Wave Motion, 45 (2007), pp. 16-29.
    • (2007) Wave Motion , vol.45 , pp. 16-29
    • Linton, C.M.1    McIver, P.2
  • 5
    • 65549169108 scopus 로고    scopus 로고
    • On the spectrum of the Dirichlet Laplacian in a narrow strip
    • L. FRIEDLANDER AND M. SOLOMYAK, On the spectrum of the Dirichlet Laplacian in a narrow strip, Israel J. Math., 170 (2009), pp. 337-354.
    • (2009) Israel J. Math. , vol.170 , pp. 337-354
    • Friedlander, L.1    Solomyak, M.2
  • 7
    • 0010627887 scopus 로고    scopus 로고
    • in English
    • J. MATH. Sci., 101 (2000), pp. 2941-2974 (in English).
    • (2000) J. Math. Sci. , vol.101 , pp. 2941-2974
  • 8
    • 11144235325 scopus 로고    scopus 로고
    • Localization effects for eigenfunctions near to the edge of a thin domain
    • S.A. NAZAROV, Localization effects for eigenfunctions near to the edge of a thin domain, Mathematica Bohemica, 127 (2002), pp. 283-292.
    • (2002) Mathematica Bohemica , vol.127 , pp. 283-292
    • Nazarov, S.A.1
  • 10
    • 0036383361 scopus 로고    scopus 로고
    • Uniform spectral asymptotics for singularly perturbed locally periodic operators
    • G. ALLAIRE AND A.L. PIATNITSKI, Uniform spectral asymptotics for singularly perturbed locally periodic operators, Comm. Partial Differential Equations, 27 (2002), pp. 705-725.
    • (2002) Comm. Partial Differential Equations , vol.27 , pp. 705-725
    • Allaire, G.1    Piatnitski, A.L.2
  • 12
    • 0023437306 scopus 로고
    • Mathematical aspects of trapping modes in the theory of water waves
    • F. URSELL, Mathematical aspects of trapping modes in the theory of water waves, J. Fluid Mech., 183 (1987), pp. 421-437.
    • (1987) J. Fluid Mech. , vol.183 , pp. 421-437
    • Ursell, F.1
  • 13
    • 0033436063 scopus 로고    scopus 로고
    • Mathematical analysis of elastic surface waves in topographic waveguides
    • A.S. BONNET-BENDHIA, J. DUTERTE, AND P. JOLY, Mathematical analysis of elastic surface waves in topographic waveguides, Math. Models Methods Appl. Sci., 9 (1999), pp. 755-798.
    • (1999) Math. Models Methods Appl. Sci. , vol.9 , pp. 755-798
    • Bonnet-Bendhia, A.S.1    Duterte, J.2    Joly, P.3
  • 14
    • 74349106401 scopus 로고    scopus 로고
    • Properties of spectra of boundary value problems in cylindrical and quasicylindrical domain, Sobolev Spaces in Mathematics II
    • V. Maz'ya, ed.
    • S.A. NAZAROV, Properties of spectra of boundary value problems in cylindrical and quasicylindrical domain, Sobolev Spaces in Mathematics II, V. Maz'ya, ed., Internat. Math. Ser., 9 (2008), pp. 261-309.
    • (2008) Internat. Math. Ser. , vol.9 , pp. 261-309
    • Nazarov, S.A.1
  • 16
    • 0013048547 scopus 로고
    • Self-adjoint extensions of the Dirichlet problem operator in weighted function spaces
    • in Russian
    • S.A. NAZAROV, Self-adjoint extensions of the Dirichlet problem operator in weighted function spaces, Mat. Sb., 137 (1988), pp. 224-241 (in Russian);
    • (1988) Mat. Sb. , vol.137 , pp. 224-241
    • Nazarov, S.A.1
  • 17
    • 0040335134 scopus 로고
    • in English
    • MATH. USSR Sb., 65 (1990), pp. 229-247 (in English).
    • (1990) Math. USSR Sb. , vol.65 , pp. 229-247
  • 20
    • 0000309125 scopus 로고
    • Boundary value problems for elliptic problems in domains with conical or corner points
    • in Russian
    • V.A. KONDRATIEV, Boundary value problems for elliptic problems in domains with conical or corner points, Trudy Moskov. Matem. Obshch., 16 (1967), pp. 209-292 (in Russian);
    • (1967) Trudy Moskov. Matem. Obshch. , vol.16 , pp. 209-292
    • Kondratiev, V.A.1
  • 21
    • 0000309125 scopus 로고
    • in English
    • TRANS. MOSCOW MATH. Soc., 16 (1967), pp. 227-313 (in English).
    • (1967) Trans. Moscow Math. Soc. , vol.16 , pp. 227-313
  • 22
    • 0000516706 scopus 로고
    • The smoothness of the solution of the Dirichlet problem for second order elliptic equations in a piecewise smooth domain
    • in Russian
    • V.A. KONDRATIEV, The smoothness of the solution of the Dirichlet problem for second order elliptic equations in a piecewise smooth domain, Differ. Uravn., 6 (1970), pp. 1831-1843 (in Russian);
    • (1970) Differ. Uravn. , vol.6 , pp. 1831-1843
    • Kondratiev, V.A.1
  • 23
    • 0039743283 scopus 로고
    • in English
    • DIFFER. EQU., 6 (1970), pp. 1392-1401 (in English).
    • (1970) Differ. Equ. , vol.6 , pp. 1392-1401
  • 24
    • 33744798864 scopus 로고    scopus 로고
    • Exponentially decreasing solutions of the problem of diffraction by a rigid periodic boundary
    • in Russian
    • I.V. KAMOTSKII AND S.A. NAZAROV, Exponentially decreasing solutions of the problem of diffraction by a rigid periodic boundary, Mat. Zametki, 73 (2003), pp. 138-140 (in Russian);
    • (2003) Mat. Zametki , vol.73 , pp. 138-140
    • Kamotskii, I.V.1    Nazarov, S.A.2
  • 25
    • 33744814195 scopus 로고    scopus 로고
    • in English
    • MATH. Notes, 73 (2003), pp. 129-131 (in English).
    • (2003) Math. Notes , vol.73 , pp. 129-131
  • 26
    • 0001561763 scopus 로고
    • Regular degeneration and boundary layer of linear differential equations with small parameter
    • M.I. VISIK AND L.A. LJUSTERNIK, Regular degeneration and boundary layer of linear differential equations with small parameter, Amer. Math. Soc. Transl., 20 (1962), pp. 239-364.
    • (1962) Amer. Math. Soc. Transl. , vol.20 , pp. 239-364
    • Visik, M.I.1    Ljusternik, L.A.2
  • 27
    • 0009297310 scopus 로고
    • Asymptotic expansions of ordinary differential equations in Hilbert space
    • A. PAZY, Asymptotic expansions of ordinary differential equations in Hilbert space, Arch. Ration. Mech. Anal., 24 (1967), pp. 193-218.
    • (1967) Arch. Ration. Mech. Anal. , vol.24 , pp. 193-218
    • Pazy, A.1
  • 28
    • 0002688101 scopus 로고
    • p and Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary
    • in Russian
    • p and Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary, Math. Nachr., 81 (1978), pp. 25-82 (in Russian);
    • (1978) Math. Nachr. , vol.81 , pp. 25-82
    • Mazja, V.G.1    Plamenevskii, B.A.2
  • 29
    • 0040767929 scopus 로고
    • in English
    • AMER. Math. Soc. Transl., 123 (1984), pp. 1-56 (in English).
    • (1984) Amer. Math. Soc. Transl. , vol.123 , pp. 1-56
  • 30
    • 0013143625 scopus 로고    scopus 로고
    • The polynomial property of self-adjoint elliptic boundary-value problems and the algebraic description of their attributes
    • in Russian
    • S.A. NAZAROV, The polynomial property of self-adjoint elliptic boundary-value problems and the algebraic description of their attributes, Uspekhi Mat. Nauk., 54 (1999), pp. 77-142 (in Russian);
    • (1999) Uspekhi Mat. Nauk. , vol.54 , pp. 77-142
    • Nazarov, S.A.1
  • 31
    • 0033262908 scopus 로고    scopus 로고
    • in English
    • RUSS. MATH. SURVEYS, 54 (1999), pp. 947-1014 (in English).
    • (1999) Russ. Math. Surveys , vol.54 , pp. 947-1014
  • 32
    • 0141546115 scopus 로고    scopus 로고
    • Asymptotic theory of thin plates and rods. Vol. 1
    • Nauchnaya Kniga, Novosibirsk
    • S.A. NAZAROV, Asymptotic Theory of Thin Plates and Rods. Vol. 1. Dimension Reduction and Integral Estimates, Nauchnaya Kniga, Novosibirsk, 2001.
    • (2001) Dimension Reduction and Integral Estimates
    • Nazarov, S.A.1
  • 33
    • 20444370062 scopus 로고    scopus 로고
    • Uniform estimates of remainders in asymptotic expansions of solutions to the problem on eigen-oscillations of a piezoelectric plate
    • in Russian
    • S.A. NAZAROV, Uniform estimates of remainders in asymptotic expansions of solutions to the problem on eigen-oscillations of a piezoelectric plate, Probl. Mat. Analiz., 25 (2003), pp. 99-188 (in Russian);
    • (2003) Probl. Mat. Analiz. , vol.25 , pp. 99-188
    • Nazarov, S.A.1
  • 34
    • 79251492923 scopus 로고    scopus 로고
    • in English
    • J. MATH. SCI. (N.Y.), 114 (2003), pp. 1657-1722 (in English).
    • (2003) J. Math. Sci. (N.Y.) , vol.114 , pp. 1657-1722
  • 39
    • 84959654606 scopus 로고
    • 2u + λu = 0 when the boundary conditions are given on semi-infinite domains
    • 2u + λu = 0 when the boundary conditions are given on semi-infinite domains, Proc. Cambridge Philos. Soc., 49 (1953), pp. 668-684.
    • (1953) Proc. Cambridge Philos. Soc. , vol.49 , pp. 668-684
    • Jones, D.S.1
  • 40
    • 20444422491 scopus 로고    scopus 로고
    • Eigen-oscillations of contrasting non-homogeneous bodies: Asymptotic and uniform estimates for eigenvalues
    • M. LOBO, S.A. NAZAROV, AND E. PEREZ, Eigen-oscillations of contrasting non-homogeneous bodies: Asymptotic and uniform estimates for eigenvalues, IMA J. Appl. Math., 70 (2005), pp. 419-458.
    • (2005) IMA J. Appl. Math. , vol.70 , pp. 419-458
    • Lobo, M.1    S.Nazarov, A.2    Perez, E.3
  • 41
    • 33644959984 scopus 로고    scopus 로고
    • Spectral stiff problems in domains surrounded by thin bands: Asymptotic and uniform estimates for eigenvalues
    • D. GOMEZ, M. LOBO, S.A. NAZAROV, AND E. PEREZ, Spectral stiff problems in domains surrounded by thin bands: Asymptotic and uniform estimates for eigenvalues, J.Math. Pures Appl., 85 (2006), pp. 598-632.
    • (2006) J.Math. Pures Appl. , vol.85 , pp. 598-632
    • Gomez, D.1    Lobo, M.2    Nazarov, S.A.3    Perez, E.4
  • 43
    • 0010664727 scopus 로고    scopus 로고
    • Mathematical elasticity. Vol. II. Theory of plates
    • North-Holland, Amsterdam
    • P.G. CIARLET, Mathematical elasticity. Vol. II. Theory of plates, Studies in Mathematics and Its Applications 27, North-Holland, Amsterdam, 1997.
    • (1997) Studies in Mathematics and its Applications , vol.27
    • Ciarlet, P.G.1
  • 44
    • 0019560197 scopus 로고
    • Two dimensional approximation of three dimensional eigenvalue problems in plate theory
    • P.G. CIARLET AND S. KESAVAN, Two dimensional approximation of three dimensional eigenvalue problems in plate theory, Comput. Methods Appl. Mech. Engrg., 26 (1981), pp. 145-172.
    • (1981) Comput. Methods Appl. Mech. Engrg. , vol.26 , pp. 145-172
    • Ciarlet, P.G.1    Kesavan, S.2
  • 45
    • 0038813791 scopus 로고    scopus 로고
    • Local problems or vibrating systems with concentrated masses: A review
    • M. LOBO AND E. PEREZ, Local problems or vibrating systems with concentrated masses: A review, C.R. Mecanique, 331 (2003), pp. 303-317.
    • (2003) C.R. Mecanique , vol.331 , pp. 303-317
    • Lobo, M.1    Perez, E.2
  • 46
    • 0028372282 scopus 로고
    • Existence theorems for trapped modes
    • D.V. EVANS, M. LEVITIN, AND D. VASIL'EV, Existence theorems for trapped modes, J. Fluid Mech., 261 (1994), pp. 21-31.
    • (1994) J. Fluid Mech. , vol.261 , pp. 21-31
    • Evans, D.V.1    Levitin, M.2    Vasil'ev, D.3
  • 47
    • 0000255676 scopus 로고
    • Interaction of concentrated masses in a harmonically oscillating spatial body with Neumann boundary conditions
    • S.A. NAZAROV, Interaction of concentrated masses in a harmonically oscillating spatial body with Neumann boundary conditions, RAIRO Model. Math. Anal. Numer., 27 (1993), pp. 777-799.
    • (1993) RAIRO Model. Math. Anal. Numer. , vol.27 , pp. 777-799
    • Nazarov, S.A.1
  • 48
    • 23044523486 scopus 로고    scopus 로고
    • On the asymptotics of the spectrum of a thin plate problem of elasticity
    • in Russian
    • S.A. NAZAROV, On the asymptotics of the spectrum of a thin plate problem of elasticity, Sib. Mat. Zh., 41 (2000), pp. 895-912 (in Russian);
    • (2000) Sib. Mat. Zh. , vol.41 , pp. 895-912
    • Nazarov, S.A.1
  • 49
    • 23044523486 scopus 로고    scopus 로고
    • in English
    • SIBERIAN MATH. J., 41 (2000), pp. 744-759 (in English).
    • (2000) Siberian Math. J. , vol.41 , pp. 744-759
  • 51
    • 84980144457 scopus 로고
    • Scattering frequencies of resonators
    • J.T. BEALE, Scattering frequencies of resonators, Commun. Pure Appl. Math., 26 (1973), pp. 549-563.
    • (1973) Commun. Pure Appl. Math. , vol.26 , pp. 549-563
    • Beale, J.T.1
  • 52
    • 77953914119 scopus 로고    scopus 로고
    • On the eigenvalues of a dumbbell with a thin handle
    • in Russian
    • R.R. GADYL'SHIN, On the eigenvalues of a dumbbell with a thin handle, Izv. Ross. Akad. Nauk, Ser. Mat., 69 (2005), pp. 45-110 (in Russian);
    • (2005) Izv. Ross. Akad. Nauk, Ser. Mat. , vol.69 , pp. 45-110
    • Gadyl'shin, R.R.1
  • 53
    • 33645416540 scopus 로고    scopus 로고
    • in English
    • IZV. MATH., 69 (2005), pp. 265-329 (in English).
    • (2005) Izv. Math. , vol.69 , pp. 265-329
  • 54
    • 84973998817 scopus 로고
    • Lower bounds on the interaction between cavities connected by a thin tube
    • R.M. BROWN, P.D. HISLOP, AND A. MARTINEZ, Lower bounds on the interaction between cavities connected by a thin tube, Duke Math. J., 73 (1994), pp. 163-176.
    • (1994) Duke Math. J. , vol.73 , pp. 163-176
    • Brown, R.M.1    Hislop, P.D.2    Martinez, A.3
  • 55
    • 0026477798 scopus 로고
    • Trapped acoustic modes
    • D.V. EVANS, Trapped acoustic modes, IMA J. Appl. Math., 49 (1992), pp. 45-60.
    • (1992) IMA J. Appl. Math. , vol.49 , pp. 45-60
    • Evans, D.V.1
  • 56
    • 74349106401 scopus 로고    scopus 로고
    • Properties of spectra of boundary value problems in cylindrical and quasicylindrical domains, Sobolev Spaces in Mathematics
    • V. Maz'ya, ed., Springer, New York
    • S.A. NAZAROV, Properties of spectra of boundary value problems in cylindrical and quasicylindrical domains, Sobolev Spaces in Mathematics. Vol. II, V. Maz'ya, ed., Internat. Math. Ser., 9 (2008), pp. 261-309, Springer, New York.
    • (2008) Internat. Math. Ser. , vol.2 , Issue.9 , pp. 261-309
    • Nazarov, S.A.1
  • 57
    • 3042686028 scopus 로고
    • Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval
    • M.A. MIRONOV, Propagation of a flexural wave in a plate whose thickness decreases smoothly to zero in a finite interval, Soviet Physics-Acoustics, 34 (1988), pp. 318-319.
    • (1988) Soviet Physics-Acoustics , vol.34 , pp. 318-319
    • Mironov, M.A.1
  • 58
    • 6344237145 scopus 로고    scopus 로고
    • New type of vibration dampers utilising the effect of acoustic "Black holes"
    • V.V. KRYLOV, New type of vibration dampers utilising the effect of acoustic "black holes", Acta Acustica united with Acustica, 90 (2004), pp. 830-837.
    • (2004) Acta Acustica united with Acustica , vol.90 , pp. 830-837
    • Krylov, V.V.1
  • 59
    • 63449109236 scopus 로고    scopus 로고
    • The "absorption" effect caused by beak-shaped boundary irregularity for elastic waves
    • in Russian
    • G. CARDONE, S.A. NAZAROV, AND J. TASKINEN, The "absorption" effect caused by beak-shaped boundary irregularity for elastic waves, Dokl. Ross. Akad. Nauk., 425 (2009), pp. 182-186 (in Russian);
    • (2009) Dokl. Ross. Akad. Nauk. , vol.425 , pp. 182-186
    • Cardone, G.1    Nazarov, S.A.2    Taskinen, J.3
  • 60
    • 63449109236 scopus 로고    scopus 로고
    • in English
    • DOKL. Phys., 54 (2009), pp. 146-150 (in English).
    • (2009) Dokl. Phys. , vol.54 , pp. 146-150
  • 61
    • 0021650084 scopus 로고
    • Perturbation of eigenvalues in thermoelasticity and vibration of systems with concentrated masses
    • E. SANCHEZ-PALENCIA, Perturbation of eigenvalues in thermoelasticity and vibration of systems with concentrated masses, Lect. Notes Physh., 195 (1984), pp. 346-368.
    • (1984) Lect. Notes Physh. , vol.195 , pp. 346-368
    • Sanchez-Palencia, E.1
  • 63
    • 35448996380 scopus 로고    scopus 로고
    • Asymptotics for the eigenelements of the Neumann spectral problem with concentrated masses
    • J. CAÍNZOS, E. PÉREZ, AND M. VILASÁNCHEZ, Asymptotics for the eigenelements of the Neumann spectral problem with concentrated masses, Indiana Univ. Math. J., 56 (2007), pp. 1939-1987.
    • (2007) Indiana Univ. Math. J. , vol.56 , pp. 1939-1987
    • Caínzos, J.1    Pérez, E.2    Vilasánchez, M.3
  • 64
    • 33645964219 scopus 로고    scopus 로고
    • On gaps in the spectrum of some elliptic operators in divergent form with periodic coefficients
    • in Russian
    • V.V. ZHIKOV, On gaps in the spectrum of some elliptic operators in divergent form with periodic coefficients, Algebra i Analiz., 16 (2004), pp. 34-58 (in Russian);
    • (2004) Algebra i Analiz. , vol.16 , pp. 34-58
    • Zhikov, V.V.1
  • 65
    • 85009754868 scopus 로고    scopus 로고
    • in English
    • ST. PETERSBG. MATH. J., 16 (2005), pp. 773-790 (in English).
    • (2005) St. Petersbg. Math. J. , vol.16 , pp. 773-790
  • 66
    • 45149112159 scopus 로고    scopus 로고
    • On the spectrum of narrow periodic waveguides
    • L. FRIEDLANDER AND M. SOLOMYAK, On the spectrum of narrow periodic waveguides, Russ. J. Math. Phys., 15 (2008), pp. 238-242.
    • (2008) Russ. J. Math. Phys. , vol.15 , pp. 238-242
    • Friedlander, L.1    Solomyak, M.2
  • 67
    • 38749098861 scopus 로고    scopus 로고
    • Extremum problems for eigenvalues of elliptic operators
    • Verlag, Basel, Birkhäuser
    • A. HENROT, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006.
    • (2006) Frontiers in Mathematics
    • Henrot, A.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.