September  2020, 28(3): 1257-1272. doi: 10.3934/era.2020069

Recent progress on the mathematical study of anomalous localized resonance in elasticity

Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China

* Corresponding author: Hongjie Li

Received  May 2020 Published  July 2020

We consider the anomalous localized resonance induced by negative elastic metamaterials and its application in invisibility cloaking. We survey the recent mathematical developments in the literature and discuss two mathematical strategies that have been developed for tackling this peculiar resonance phenomenon. The first one is the spectral method, which explores the anomalous localized resonance through investigating the spectral system of the associated Neumann-Poincaré operator. The other one is the variational method, which considers the anomalous localized resonance via calculating the nontrivial kernels of a non-elliptic partial differential operator. The advantages and the relationship between the two methods are discussed. Finally, we propose some open problems for the future study.

Citation: Hongjie Li. Recent progress on the mathematical study of anomalous localized resonance in elasticity. Electronic Research Archive, 2020, 28 (3) : 1257-1272. doi: 10.3934/era.2020069
References:
[1]

H. Ammari, G. Ciraolo, H. Kang, H. Lee and G. W. Milton, Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance II, in Inverse Problems and Applications, Contemp. Math., 615, Amer. Math. Soc., Providence, RI, (2014), 1–14.  Google Scholar

[2]

H. Ammari, G. Ciraolo, H. Kang, H. Lee and G. W. Milton, Anomalous localized resonance using a folded geometry in three dimensions, Proc. R. Soc. A, 469 (2013), 20130048. doi: 10.1098/rspa.2013.0048.  Google Scholar

[3]

H. AmmariG. CiraoloH. KangH. Lee and G. W. Milton, Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance, Arch. Ration. Mech. Anal., 208 (2013), 667-692.  doi: 10.1007/s00205-012-0605-5.  Google Scholar

[4]

H. AmmariY. Deng and P. Millien, Surface plasmon resonance of nanoparticles and applications in imaging, Arch. Ration. Mech. Anal., 220 (2016), 109-153.  doi: 10.1007/s00205-015-0928-0.  Google Scholar

[5]

H. AmmariB. FitzpatrickD. GontierH. Lee and H. Zhang, Minnaert resonances for acoustic waves in bubbly media, Ann. Inst. H. Poincaré Anal. Non Linéare, 35 (2018), 1975-1998.  doi: 10.1016/j.anihpc.2018.03.007.  Google Scholar

[6]

H. AmmariP. MillienM. Ruiz and H. Zhang, Mathematical analysis of plasmonic nanoparticles: The scalar case, Arch. Ration. Mech. Anal., 224 (2017), 597-658.  doi: 10.1007/s00205-017-1084-5.  Google Scholar

[7]

H. AmmariM. RuizS. Yu and H. Zhang, Mathematical analysis of plasmonic resonances for nanoparticles: The full Maxwell equations, J. Differential Equations, 261 (2016), 3615-3669.  doi: 10.1016/j.jde.2016.05.036.  Google Scholar

[8]

K. AndoY.-G. JiH. KangK. Kim and S. Yu, Spectral properties of the Neumann-Poincaré operator and cloaking by anomalous localized resonance for the elasto-static system, European J. Appl. Math., 29 (2018), 189-225.  doi: 10.1017/S0956792517000080.  Google Scholar

[9]

K. Ando and H. Kang, Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operator, J. Math. Anal. Appl., 435 (2016), 162-178.  doi: 10.1016/j.jmaa.2015.10.033.  Google Scholar

[10]

K. Ando, H. Kang, K. Kim and S. Yu, Cloaking by anomalous localized resonance for linear elasticity on a coated structure, preprint, arXiv: 1612.08384. Google Scholar

[11]

K. AndoH. Kang and H. Liu, Plasmon resonance with finite frequencies: A validation of the quasi-static approximation for diametrically small inclusions, SIAM J. Appl. Math., 76 (2016), 731-749.  doi: 10.1137/15M1025943.  Google Scholar

[12]

K. AndoH. Kang and Y. Miyanishi, Elastic Neumann-Poincaré operators on three dimensional smooth domains: Polynomial compactness and spectral structure, Int. Math. Res. Not. IMRN, 2019 (2019), 3883-3900.  doi: 10.1093/imrn/rnx258.  Google Scholar

[13]

E. BlåstenH. LiH. Liu and Y. Wang, Localization and geometrization in plasmon resonances and geometric structures of Neumann-Poincaré eigenfunctions, ESAIM Math. Model. Numer. Anal., 54 (2020), 957-976.  doi: 10.1051/m2an/2019091.  Google Scholar

[14]

G. Bouchitté and B. Schweizer, Cloaking of small objects by anomalous localized resonance, Quart. J. Mech. Appl. Math., 63 (2010), 437-463.  doi: 10.1093/qjmam/hbq008.  Google Scholar

[15]

O. P. Bruno and S. Lintner, Superlens-cloaking of small dielectric bodies in the quasistatic regime, J. Appl. Phys., 102 (2007). doi: 10.1063/1.2821759.  Google Scholar

[16]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar

[17]

Y. DengH. Li and H. Liu, On spectral properties of Neumann-Poincaré operator and plasmonic cloaking in 3D elastostatics, J. Spectr. Theory, 9 (2019), 767-789.  doi: 10.4171/JST/262.  Google Scholar

[18]

Y. DengH. Li and H. Liu, Analysis of surface polariton resonance for nanoparticles in elastic system, SIAM J. Math. Anal., 52 (2020), 1786-1805.  doi: 10.1137/18M1181067.  Google Scholar

[19]

Y. Deng, H. Li and H. Liu, Spectral properties of Neumann-Poincaré operator and anomalous localized resonance in elasticity beyond quasi-static limit, J. Elasticity, (2020). doi: 10.1007/s10659-020-09767-8.  Google Scholar

[20]

H. KettunenM. Lassas and P. Ola, On absence and existence of the anomalous localized resonance without the quasi-static approximation, SIAM J. Appl. Math., 78 (2018), 609-628.  doi: 10.1137/16M1097055.  Google Scholar

[21]

R. V. KohnJ. LuB. Schweizer and M. I. Weinstein, A variational perspective on cloaking by anomalous localized resonance, Comm. Math. Phys., 328 (2014), 1-27.  doi: 10.1007/s00220-014-1943-y.  Google Scholar

[22]

V. D. Kupradze, T. G. Gegelia, M. O. Bashele${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$shvili and T. V. Burchuladze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland Series in Applied Mathematics and Mechanics, 25, North-Holland Publishing Co., Amsterdam-New York, 1979.  Google Scholar

[23]

J. Li and C. T. Chan, Double-negative acoustic metamaterial, Phys. Rev. E, 70 (2004). doi: 10.1103/PhysRevE.70.055602.  Google Scholar

[24]

H. Li, J. Li and H. Liu, On novel elastic structures inducing polariton resonances with finite frequencies and cloaking due to anomalous localized resonances, J. Math. Pures Appl. (9), 120 (2018), 195–219. doi: 10.1016/j.matpur.2018.06.014.  Google Scholar

[25]

H. LiJ. Li and H. Liu, On quasi-static cloaking due to anomalous localized resonance in $\mathbb{R}^3$, SIAM J. Appl. Math., 75 (2015), 1245-1260.  doi: 10.1137/15M1009974.  Google Scholar

[26]

H. Li, S. Li, H. Liu and X. Wang, Analysis of electromagnetic scattering from plasmonic inclusions beyond the quasi-static approximation and applications, ESAIM: Math. Model. Numer. Anal., 53 (2019), 1351–1371. doi: 10.1051/m2an/2019004.  Google Scholar

[27]

H. Li and H. Liu, On anomalous localized resonance for the elastostatic system, SIAM J. Math. Anal., 48 (2016), 3322-3344.  doi: 10.1137/16M1059023.  Google Scholar

[28]

H. Li and H. Liu, On three-dimensional plasmon resonances in elastostatics, Ann. Mat. Pura Appl. (4), 196 (2017), 1113–1135. doi: 10.1007/s10231-016-0609-0.  Google Scholar

[29]

H. Li and H. Liu, On anomalous localized resonance and plasmonic cloaking beyond the quasistatic limit, Proc. Roy. Soc. A, 474 (2018). doi: 10.1098/rspa.2018.0165.  Google Scholar

[30]

H. Li, H. Liu and J. Zou, Minnaert resonances for bubbles in soft elastic materials, preprint, arXiv: 1911.03718. Google Scholar

[31]

R. C. McPhedranN.-A. P. NicoroviciL. C. Botten and G. W. Milton, Cloaking by plasmonic resonance among systems of particles: Cooperation or combat?, C. R. Phys., 10 (2009), 391-399.  doi: 10.1016/j.crhy.2009.03.007.  Google Scholar

[32]

G. W. Milton and N.-A. P. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 3027-3059.  doi: 10.1098/rspa.2006.1715.  Google Scholar

[33]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko and Z. Jacob, Solutions in folded geometries, and associated cloaking due to anomalous resonance, New. J. Phys., 10 (2008). doi: 10.1088/1367-2630/10/11/115021.  Google Scholar

[34]

G. W. MiltonN.-A. P. NicoroviciR. C. McPhedran and V. A. Podolskiy, A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 3999-4034.  doi: 10.1098/rspa.2005.1570.  Google Scholar

[35]

J. C. Nédélec, Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems, Applied Mathematical Sciences, 144, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4393-7.  Google Scholar

[36]

N.-A. P. Nicorovici, R. C. McPhedran, S. Enoch and G. Tayeb, Finite wavelength cloaking by plasmonic resonance, New. J. Phys., 10 (2008). doi: 10.1088/1367-2630/10/11/115020.  Google Scholar

[37]

N.-A. P. NicoroviciR. C. McPhedran and G. W. Milton, Optical and dielectric properties of partially resonant composites, Phys. Rev. B, 49 (1994), 8479-8482.  doi: 10.1103/PhysRevB.49.8479.  Google Scholar

[38]

N.-A. P. NicoroviciG. W. MiltonR. C. McPhedran and L. C. Botten, Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance, Optics Express, 15 (2007), 6314-6323.  doi: 10.1364/OE.15.006314.  Google Scholar

[39]

J. B. PendryA. J. HoldenD. J. Robbins and W. J. Stewart, Low frequency plasmons in thin-wire structures, J. Phys. Condens. Matter, 10 (1998), 4785-4809.  doi: 10.1088/0953-8984/10/22/007.  Google Scholar

[40]

J. B. PendryA. J. HoldenD. J. Robbins and W. J. Stewart, Magnetism from conductors and enhanced nonlinear phenomena, IEEE Trans. Microwave Theory Techniques, 47 (1999), 2075-2084.  doi: 10.1109/22.798002.  Google Scholar

[41]

D. R. SmithW. J. PadillaD. C. VierS. C. Nemat-Nasser and S. Schultz, Composite medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett., 84 (2000), 4184-4187.  doi: 10.1103/PhysRevLett.84.4184.  Google Scholar

[42]

V. G. Veselago, The electrodynamics of substances with simultaneously negative values of $\epsilon$ and $\mu$, Sov. Phys. Usp., 10 (1968). doi: 10.1070/PU1968v010n04ABEH003699.  Google Scholar

[43]

Y. Wu, Y. Lai and Z.-Q. Zhang, Effective medium theory for elastic metamaterials in two dimensions, Phys. Rev. B, 76 (2007). doi: 10.1103/PhysRevB.76.205313.  Google Scholar

show all references

References:
[1]

H. Ammari, G. Ciraolo, H. Kang, H. Lee and G. W. Milton, Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance II, in Inverse Problems and Applications, Contemp. Math., 615, Amer. Math. Soc., Providence, RI, (2014), 1–14.  Google Scholar

[2]

H. Ammari, G. Ciraolo, H. Kang, H. Lee and G. W. Milton, Anomalous localized resonance using a folded geometry in three dimensions, Proc. R. Soc. A, 469 (2013), 20130048. doi: 10.1098/rspa.2013.0048.  Google Scholar

[3]

H. AmmariG. CiraoloH. KangH. Lee and G. W. Milton, Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance, Arch. Ration. Mech. Anal., 208 (2013), 667-692.  doi: 10.1007/s00205-012-0605-5.  Google Scholar

[4]

H. AmmariY. Deng and P. Millien, Surface plasmon resonance of nanoparticles and applications in imaging, Arch. Ration. Mech. Anal., 220 (2016), 109-153.  doi: 10.1007/s00205-015-0928-0.  Google Scholar

[5]

H. AmmariB. FitzpatrickD. GontierH. Lee and H. Zhang, Minnaert resonances for acoustic waves in bubbly media, Ann. Inst. H. Poincaré Anal. Non Linéare, 35 (2018), 1975-1998.  doi: 10.1016/j.anihpc.2018.03.007.  Google Scholar

[6]

H. AmmariP. MillienM. Ruiz and H. Zhang, Mathematical analysis of plasmonic nanoparticles: The scalar case, Arch. Ration. Mech. Anal., 224 (2017), 597-658.  doi: 10.1007/s00205-017-1084-5.  Google Scholar

[7]

H. AmmariM. RuizS. Yu and H. Zhang, Mathematical analysis of plasmonic resonances for nanoparticles: The full Maxwell equations, J. Differential Equations, 261 (2016), 3615-3669.  doi: 10.1016/j.jde.2016.05.036.  Google Scholar

[8]

K. AndoY.-G. JiH. KangK. Kim and S. Yu, Spectral properties of the Neumann-Poincaré operator and cloaking by anomalous localized resonance for the elasto-static system, European J. Appl. Math., 29 (2018), 189-225.  doi: 10.1017/S0956792517000080.  Google Scholar

[9]

K. Ando and H. Kang, Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operator, J. Math. Anal. Appl., 435 (2016), 162-178.  doi: 10.1016/j.jmaa.2015.10.033.  Google Scholar

[10]

K. Ando, H. Kang, K. Kim and S. Yu, Cloaking by anomalous localized resonance for linear elasticity on a coated structure, preprint, arXiv: 1612.08384. Google Scholar

[11]

K. AndoH. Kang and H. Liu, Plasmon resonance with finite frequencies: A validation of the quasi-static approximation for diametrically small inclusions, SIAM J. Appl. Math., 76 (2016), 731-749.  doi: 10.1137/15M1025943.  Google Scholar

[12]

K. AndoH. Kang and Y. Miyanishi, Elastic Neumann-Poincaré operators on three dimensional smooth domains: Polynomial compactness and spectral structure, Int. Math. Res. Not. IMRN, 2019 (2019), 3883-3900.  doi: 10.1093/imrn/rnx258.  Google Scholar

[13]

E. BlåstenH. LiH. Liu and Y. Wang, Localization and geometrization in plasmon resonances and geometric structures of Neumann-Poincaré eigenfunctions, ESAIM Math. Model. Numer. Anal., 54 (2020), 957-976.  doi: 10.1051/m2an/2019091.  Google Scholar

[14]

G. Bouchitté and B. Schweizer, Cloaking of small objects by anomalous localized resonance, Quart. J. Mech. Appl. Math., 63 (2010), 437-463.  doi: 10.1093/qjmam/hbq008.  Google Scholar

[15]

O. P. Bruno and S. Lintner, Superlens-cloaking of small dielectric bodies in the quasistatic regime, J. Appl. Phys., 102 (2007). doi: 10.1063/1.2821759.  Google Scholar

[16]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar

[17]

Y. DengH. Li and H. Liu, On spectral properties of Neumann-Poincaré operator and plasmonic cloaking in 3D elastostatics, J. Spectr. Theory, 9 (2019), 767-789.  doi: 10.4171/JST/262.  Google Scholar

[18]

Y. DengH. Li and H. Liu, Analysis of surface polariton resonance for nanoparticles in elastic system, SIAM J. Math. Anal., 52 (2020), 1786-1805.  doi: 10.1137/18M1181067.  Google Scholar

[19]

Y. Deng, H. Li and H. Liu, Spectral properties of Neumann-Poincaré operator and anomalous localized resonance in elasticity beyond quasi-static limit, J. Elasticity, (2020). doi: 10.1007/s10659-020-09767-8.  Google Scholar

[20]

H. KettunenM. Lassas and P. Ola, On absence and existence of the anomalous localized resonance without the quasi-static approximation, SIAM J. Appl. Math., 78 (2018), 609-628.  doi: 10.1137/16M1097055.  Google Scholar

[21]

R. V. KohnJ. LuB. Schweizer and M. I. Weinstein, A variational perspective on cloaking by anomalous localized resonance, Comm. Math. Phys., 328 (2014), 1-27.  doi: 10.1007/s00220-014-1943-y.  Google Scholar

[22]

V. D. Kupradze, T. G. Gegelia, M. O. Bashele${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$shvili and T. V. Burchuladze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland Series in Applied Mathematics and Mechanics, 25, North-Holland Publishing Co., Amsterdam-New York, 1979.  Google Scholar

[23]

J. Li and C. T. Chan, Double-negative acoustic metamaterial, Phys. Rev. E, 70 (2004). doi: 10.1103/PhysRevE.70.055602.  Google Scholar

[24]

H. Li, J. Li and H. Liu, On novel elastic structures inducing polariton resonances with finite frequencies and cloaking due to anomalous localized resonances, J. Math. Pures Appl. (9), 120 (2018), 195–219. doi: 10.1016/j.matpur.2018.06.014.  Google Scholar

[25]

H. LiJ. Li and H. Liu, On quasi-static cloaking due to anomalous localized resonance in $\mathbb{R}^3$, SIAM J. Appl. Math., 75 (2015), 1245-1260.  doi: 10.1137/15M1009974.  Google Scholar

[26]

H. Li, S. Li, H. Liu and X. Wang, Analysis of electromagnetic scattering from plasmonic inclusions beyond the quasi-static approximation and applications, ESAIM: Math. Model. Numer. Anal., 53 (2019), 1351–1371. doi: 10.1051/m2an/2019004.  Google Scholar

[27]

H. Li and H. Liu, On anomalous localized resonance for the elastostatic system, SIAM J. Math. Anal., 48 (2016), 3322-3344.  doi: 10.1137/16M1059023.  Google Scholar

[28]

H. Li and H. Liu, On three-dimensional plasmon resonances in elastostatics, Ann. Mat. Pura Appl. (4), 196 (2017), 1113–1135. doi: 10.1007/s10231-016-0609-0.  Google Scholar

[29]

H. Li and H. Liu, On anomalous localized resonance and plasmonic cloaking beyond the quasistatic limit, Proc. Roy. Soc. A, 474 (2018). doi: 10.1098/rspa.2018.0165.  Google Scholar

[30]

H. Li, H. Liu and J. Zou, Minnaert resonances for bubbles in soft elastic materials, preprint, arXiv: 1911.03718. Google Scholar

[31]

R. C. McPhedranN.-A. P. NicoroviciL. C. Botten and G. W. Milton, Cloaking by plasmonic resonance among systems of particles: Cooperation or combat?, C. R. Phys., 10 (2009), 391-399.  doi: 10.1016/j.crhy.2009.03.007.  Google Scholar

[32]

G. W. Milton and N.-A. P. Nicorovici, On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 3027-3059.  doi: 10.1098/rspa.2006.1715.  Google Scholar

[33]

G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. Cherednichenko and Z. Jacob, Solutions in folded geometries, and associated cloaking due to anomalous resonance, New. J. Phys., 10 (2008). doi: 10.1088/1367-2630/10/11/115021.  Google Scholar

[34]

G. W. MiltonN.-A. P. NicoroviciR. C. McPhedran and V. A. Podolskiy, A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 3999-4034.  doi: 10.1098/rspa.2005.1570.  Google Scholar

[35]

J. C. Nédélec, Acoustic and Electromagnetic Equations. Integral Representations for Harmonic Problems, Applied Mathematical Sciences, 144, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4393-7.  Google Scholar

[36]

N.-A. P. Nicorovici, R. C. McPhedran, S. Enoch and G. Tayeb, Finite wavelength cloaking by plasmonic resonance, New. J. Phys., 10 (2008). doi: 10.1088/1367-2630/10/11/115020.  Google Scholar

[37]

N.-A. P. NicoroviciR. C. McPhedran and G. W. Milton, Optical and dielectric properties of partially resonant composites, Phys. Rev. B, 49 (1994), 8479-8482.  doi: 10.1103/PhysRevB.49.8479.  Google Scholar

[38]

N.-A. P. NicoroviciG. W. MiltonR. C. McPhedran and L. C. Botten, Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance, Optics Express, 15 (2007), 6314-6323.  doi: 10.1364/OE.15.006314.  Google Scholar

[39]

J. B. PendryA. J. HoldenD. J. Robbins and W. J. Stewart, Low frequency plasmons in thin-wire structures, J. Phys. Condens. Matter, 10 (1998), 4785-4809.  doi: 10.1088/0953-8984/10/22/007.  Google Scholar

[40]

J. B. PendryA. J. HoldenD. J. Robbins and W. J. Stewart, Magnetism from conductors and enhanced nonlinear phenomena, IEEE Trans. Microwave Theory Techniques, 47 (1999), 2075-2084.  doi: 10.1109/22.798002.  Google Scholar

[41]

D. R. SmithW. J. PadillaD. C. VierS. C. Nemat-Nasser and S. Schultz, Composite medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett., 84 (2000), 4184-4187.  doi: 10.1103/PhysRevLett.84.4184.  Google Scholar

[42]

V. G. Veselago, The electrodynamics of substances with simultaneously negative values of $\epsilon$ and $\mu$, Sov. Phys. Usp., 10 (1968). doi: 10.1070/PU1968v010n04ABEH003699.  Google Scholar

[43]

Y. Wu, Y. Lai and Z.-Q. Zhang, Effective medium theory for elastic metamaterials in two dimensions, Phys. Rev. B, 76 (2007). doi: 10.1103/PhysRevB.76.205313.  Google Scholar

[1]

Kevin Arfi, Anna Rozanova-Pierrat. Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 1-26. doi: 10.3934/dcdss.2019001

[2]

Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1

[3]

D. Bonheure, C. Fabry. A variational approach to resonance for asymmetric oscillators. Communications on Pure & Applied Analysis, 2007, 6 (1) : 163-181. doi: 10.3934/cpaa.2007.6.163

[4]

O. A. Veliev. Essential spectral singularities and the spectral expansion for the Hill operator. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2227-2251. doi: 10.3934/cpaa.2017110

[5]

Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Neumann equations driven by a nonhomogeneous differential operator. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1055-1078. doi: 10.3934/cpaa.2011.10.1055

[6]

Eduardo Lara, Rodolfo Rodríguez, Pablo Venegas. Spectral approximation of the curl operator in multiply connected domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 235-253. doi: 10.3934/dcdss.2016.9.235

[7]

Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665

[8]

Mario Ahues, Filomena D. d'Almeida, Alain Largillier, Paulo B. Vasconcelos. Defect correction for spectral computations for a singular integral operator. Communications on Pure & Applied Analysis, 2006, 5 (2) : 241-250. doi: 10.3934/cpaa.2006.5.241

[9]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013

[10]

Sergey V. Bolotin, Piero Negrini. Variational approach to second species periodic solutions of Poincaré of the 3 body problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1009-1032. doi: 10.3934/dcds.2013.33.1009

[11]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036

[12]

Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043

[13]

Laurent Amour, Jérémy Faupin. Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials. Inverse Problems & Imaging, 2013, 7 (4) : 1115-1122. doi: 10.3934/ipi.2013.7.1115

[14]

Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013

[15]

G. M. de Araújo, S. B. de Menezes. On a variational inequality for the Navier-Stokes operator with variable viscosity. Communications on Pure & Applied Analysis, 2006, 5 (3) : 583-596. doi: 10.3934/cpaa.2006.5.583

[16]

Shitao Liu, Roberto Triggiani. Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5217-5252. doi: 10.3934/dcds.2013.33.5217

[17]

Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264

[18]

Yoshitsugu Kabeya. Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3529-3559. doi: 10.3934/dcds.2020040

[19]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Stability for determination of Riemannian metrics by spectral data and Dirichlet-to-Neumann map limited on arbitrary subboundary. Inverse Problems & Imaging, 2019, 13 (6) : 1213-1258. doi: 10.3934/ipi.2019054

[20]

Zhong-Qing Wang, Jing-Xia Wu. Generalized Jacobi rational spectral methods with essential imposition of Neumann boundary conditions in unbounded domains. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 325-346. doi: 10.3934/dcdsb.2012.17.325

2018 Impact Factor: 0.263

Metrics

  • PDF downloads (30)
  • HTML views (113)
  • Cited by (0)

Other articles
by authors

[Back to Top]