The band-gap structure of the spectrum in a periodic medium of masonry type

Günter Leugering; Sergei A. Nazarov; Jari Taskinen; Günter Leugering; Sergei A. Nazarov; Jari Taskinen

doi:10.3934/nhm.2020014

Networks and Heterogeneous Media

2020, Volume 15, Issue 4: 555-580. doi: 10.3934/nhm.2020014

Previous Article Next Article

The band-gap structure of the spectrum in a periodic medium of masonry type

1.
Department Mathematik, Lehrstuhl für Angewandte Mathematik 2, Cauerstr. 11, 91058 Erlangen, Germany
2.
Saint-Petersburg State University, Universitetskaya nab. 7–9, St. Petersburg, 199034, Russia, and, Institute for Problems in Mechanical Engineering of RAS, St. Petersburg, 199178, Russia
3.
Department of Mathematics and Statistics, University of Helsinki, P.O.Box 68, 00014 Helsinki, Finland

Received: 01 August 2019 Revised: 01 April 2020 Published: 12 July 2020
Primary: 35J57; Secondary: 35P99, 47B25

We consider the spectrum of a class of positive, second-order elliptic systems of partial differential equations defined in the plane $\mathbb{R}^2$ . The coefficients of the equation are assumed to have a special form, namely, they are doubly periodic and of high contrast. More precisely, the plane $\mathbb{R}^2$ is decomposed into an infinite union of the translates of the rectangular periodicity cell $\Omega^0$ , and this in turn is divided into two components, on each of which the coefficients have different, constant values. Moreover, the second component of $\Omega^0$ consist of a neighborhood of the boundary of the cell of the width $h$ and thus has an area comparable to $h$ , where $h>0$ is a small parameter.

Using the methods of asymptotic analysis we study the position of the spectral bands as $h \to 0$ and in particular show that the spectrum has at least a given, arbitrarily large number of gaps, provided $h$ is small enough.

Keywords:

Citation: Günter Leugering, Sergei A. Nazarov, Jari Taskinen. The band-gap structure of the spectrum in a periodic medium of masonry type[J]. Networks and Heterogeneous Media, 2020, 15(4): 555-580. doi: 10.3934/nhm.2020014

Related Papers:

[1]	Charles Bordenave, David R. McDonald, Alexandre Proutière . A particle system in interaction with a rapidly varying environment: Mean field limits and applications. Networks and Heterogeneous Media, 2010, 5(1): 31-62. doi: 10.3934/nhm.2010.5.31
[2]	Michele Gianfelice, Enza Orlandi . Dynamics and kinetic limit for a system of noiseless $d$-dimensional Vicsek-type particles. Networks and Heterogeneous Media, 2014, 9(2): 269-297. doi: 10.3934/nhm.2014.9.269
[3]	Fabio Camilli, Italo Capuzzo Dolcetta, Maurizio Falcone . Preface. Networks and Heterogeneous Media, 2012, 7(2): i-ii. doi: 10.3934/nhm.2012.7.2i
[4]	Peter V. Gordon, Cyrill B. Muratov . Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks and Heterogeneous Media, 2012, 7(4): 767-780. doi: 10.3934/nhm.2012.7.767
[5]	Maria Teresa Chiri, Xiaoqian Gong, Benedetto Piccoli . Mean-field limit of a hybrid system for multi-lane car-truck traffic. Networks and Heterogeneous Media, 2023, 18(2): 723-752. doi: 10.3934/nhm.2023031
[6]	Olivier Guéant . New numerical methods for mean field games with quadratic costs. Networks and Heterogeneous Media, 2012, 7(2): 315-336. doi: 10.3934/nhm.2012.7.315
[7]	Michael Herty, Lorenzo Pareschi, Giuseppe Visconti . Mean field models for large data–clustering problems. Networks and Heterogeneous Media, 2020, 15(3): 463-487. doi: 10.3934/nhm.2020027
[8]	Michael Herty, Lorenzo Pareschi, Sonja Steffensen . Mean--field control and Riccati equations. Networks and Heterogeneous Media, 2015, 10(3): 699-715. doi: 10.3934/nhm.2015.10.699
[9]	Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang . Uniform stability and mean-field limit for the augmented Kuramoto model. Networks and Heterogeneous Media, 2018, 13(2): 297-322. doi: 10.3934/nhm.2018013
[10]	Martino Bardi . Explicit solutions of some linear-quadratic mean field games. Networks and Heterogeneous Media, 2012, 7(2): 243-261. doi: 10.3934/nhm.2012.7.243

Abstract

1. The Journal

Networks and Heterogeneous Media (NHM) was founded in 2006 and has been growing successfully almost for 20 years. Responding to the journal's needs, NHM began its transformation at the end of 2022, officially changing to an OA publishing model in 2023 for the first time. From the start of the new submission system in August 2022 until December 20, 2023, the journal received a total of 330 submissions, and 80 were online, with a rejection rate of 73%, which shows that, despite the increase in publication, NHM has always maintained high standards and strict requirements. This would not have been possible without the support of our editor-in-chief and editorial board team. In the meantime, thanks to the whole EB for the work done, our editorial board has been enlarged this year with the inclusion of some outstanding young scholars. Next, journal development, manuscript processing, and future perspectives will be presented to share NHM's work and development this year.

Submission	Online	Reject/Withdraw
Data source from August 01, 2022–December 20, 2023.
330	80	217/25

2. Manuscript processing

Here you will find the processing time for each stage of the paper, the turnaround time for publication, and the national & regional statistics of the authors.

2.1. Manuscript processing time

The processing time of the manuscript comprises three measurement indicators: Average Publication Time (APT), Submission to First Decision Time (TFD), and Acceptance to Publication Time (ATOP). Each indicator includes annual average time and quarterly time.

1. APT

In the figure, the horizontal axis represents the quarter-year, the vertical axis represents the number of days, and the bar graph represents the average value of APT for each quarter. The red line indicates the annual average ATP for the year 2022, while the green line represents the same for 2023.

Since August 2023, the review period for NHM has been prolonged from 14 days to 30 days. Consequently, it is evident that starting from the second quarter of 2023, the publication cycle of NHM has progressively extended from approximately 2.6 months to 4.8 months. Ideally, the Average Publication Time (APT) for a manuscript in an Open Access (OA) journal is around 60 days.

2. TFD

TFD is the time from receipt of the manuscript to the first decision, including the time for the editorial board to do a brief check and the reviewers to review the manuscript. The average TFD for 2023 is 58.75 days. It is worth noting that the editors also waited for reviewers for much longer than 14 days when the required review period was 14 days. Similarly, after the required review time of 30 days, editors waited much longer than 30 days for reviewers, and in some holiday months, such as Christmas, it even went to 45–60 days.

3. ATOP

This section shows the average time from manuscript acceptance to publication, usually 10 days, which is influenced by the typesetting editor, the English editor, and the author's cooperation. The average ATOP for 2023 is 19.34 days.

2.2. Distribution of countries

This section presents the geographic distribution of submitted manuscripts and published manuscripts. The distribution of author groups, serving as an indicator of a journal's future focus, constitutes a broad and influential category. This strong group has the potential to enhance the journal's citation impact, fostering its growth and prosperity.

This view provides the counts of Submitted manuscripts per region and country. The region and country are derived by the affiliation of the author. The top 10 countries list is computed using Submitted articles descending for 2023.

Submissions to NHM are mostly from countries in Asia, such as China, Saudi Arabia, Pakistan, etc.; final publications are mostly from countries in Asia and Europe, such as China, India, France, Italy, etc.

3. Journal development

3.1. Editorial board countries and regions

Currently, NHM has 56 editorial board members from 14 countries on five continents, with the highest number of editorial board members from Europe, followed by Asia and North America.

This year we have joined six new members of the Editorial Board, whose information is shown in the table below. We welcome them and hope to attract more outstanding scholars to join our team.

Name	Position	Affiliation
Xian-Ming Gu	Associate Professor	School of Mathematics, Southwestern University of Finance and Economics (SWUFE), Chengdu, China
Dante Kalise	Senior Lecturer	Department of Mathematics, AMMP Section Imperial College London, UK
Emiliano Cristiani	Professor	Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche, Rome, Italy
Eduardo Casas Renteria	Professor	Department of Applied Mathematics and Computer Science, E.T.S.I. Industriales and Telecommunication, University of Cantabria, Santander, Spain
Giuseppe Maria Coclite	Professor	Department of Mechanics, Mathematics and Management Polytechnic University of Bari, Italy
Delio Mugnolo	Professor	Faculty of Mathematics and Computer Science University of Hagen, Hagen, Germany

3.2. Article metrics

Statistics of the most cited manuscripts of Web of Science in the last five years and the last two years, where the "*" after the title indicates that the manuscript is from a special issue manuscript.

1. Most Cited Articles, 2023 (Last Five Years)

Title	Authors	Publication Year	Total Citations	Average per Year
Note: "*" Stands for Contributions to the Special Issue. Last Updated: December 2023 Source: Web of Science
Multiscale models of COVID-19 with mutations and variants*	Nicola Bellomo, Diletta Burini, Nisrine Outada	2022	29	14.5
Emergent behavior of cucker-smale model with normalized weights and distributed time delays	Young-Pil Choi, Cristina Pignotti	2019	21	4.2
Non-local multi-class traffic flow models	Felisia Angela Chiarello, Paola Goatin	2019	17	3.4
Opinion formation in voting processes under bounded confidence	Sergei Yu Pilyugin, M. C. Campi	2019	16	3.2
Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations*	Luis Almeida, Federica Bubba, Benoit Perthame, Camille Pouchol	2019	16	3.2
Deep neural network approach to forward-inverse problems	Hyeontae Jo, Hwijae Son, Hyung Ju Hwang, Eun Heui Kim	2020	13	3.25
Relative entropy method for the relaxation limit of hydrodynamic models*	Jose Antonio Carrillo, Yingping Peng, Aneta Wroblewska-Kaminska	2020	11	2.75
Homogenization of Bingham flow in thin porous media	Maria Anguiano, Renata Bunoiu	2020	11	2.75
Incompressible limit of a continuum model of tissue growth for two cell populations	Pierre Degond, Sophie Hecht, Nicolas Vauchelet	2020	11	2.75
Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge: a study of fractional calculus on metric graph	Vaibhav Mehandiratta, Mani Mehra, Guenter Leugering	2021	10	3.33

2. Most Cited Articles, 2023 (Last Two Years)

Title	Authors	Publication Year	Total Citations	Average per Year
Note: "*" Stands for Contributions to the Special Issue. Last Updated: December 2023 Source: Web of Science
Multiscale models of COVID-19 with mutations and variants*	Nicola Bellomo, Diletta Burini, Nisrine Outada	2022	29	14.5
Compactness property of the linearized Boltzmann operator for a diatomic single gas model	Stephane Brull, Marwa Shahine, Philippe Thieullen	2022	5	2.5
An sir-like kinetic model tracking individuals' viral load*	Rossella Della Marca, Nadia Loy, Andrea Tosin	2022	5	2.5
A study of computational and conceptual complexities of compartment and agent based models*	Prateek Kunwar, Oleksandr Markovichenko, Monique Chyba, Yuriy Mileyko, Alice Koniges, Thomas Lee	2022	5	2.5
Homogenization of nonlinear nonlocal diffusion equation with periodic and stationary structure	Junlong Chen, Yanbin Tang	2023	3	3
Global solution to the Cauchy problem of fractional drift diffusion system with power-law nonlinearity	Caihong Gu, Yanbin Tang	2023	3	3
Vaccination strategies through intra-compartmental dynamics*	Rinaldo M. Colombo, Francesca Marcellini, Elena Rossi	2022	3	1.5
A measure model for the spread of viral infections with mutations*	Xiaoqian Gong, Benedetto Piccoli	2022	3	1.5
Optimization of vaccination for COVID-19 in the midst of a pandemic*	Qi Luo, Ryan Weightman, Sean T. McQuade, Mateo Diaz, Emmanuel Trelat, William Barbour, Dan Work, Samitha Samaranayake, Benedetto Piccoli	2022	3	1.5
Asymptotic flocking of the relativistic Cucker-Smale model with time delay	Hyunjin Ahn	2023	2	2

3.3. Special issues

Only the number of submissions and rejections, publications for the special issue were counted from August 1, 2022, to December 20, 2023.

Special Issue Submissions	Rejection and withdrawal	Published
136	53/12	60

The data counts the submissions, rejections, and published manuscripts for special issues established in 2023.

Title	Established	Contribute	Accept	Reject
Recent advances in numerical methods for integer-and fractional-order PDEs	2022-08-23	47	29	18
Nonlocal conservation laws	2022-08-24	7	5	1
New trends on discrete networks	2022-09-27	31	12	15
Traffic and autonomy	2023-01-11	7	7	0
Advanced Mathematical Methodologies to Manage Pandemics	2023-05-04	9	2	3
Interdisciplinary Approaches for Understanding Networks and Heterogeneous Media	2023-05-09	17	5	11
Numerical Simulation and Mathematical Modelling	2023-11-03	14	0	6
Nonlinear PDEs in material science	2023-11-16	0	0	0
Analysis of Analytical, Computational and ML-based Approaches for Differential and Integral Equations	2023-11-29	0	0	0

4. Summary and perspective

4.1. Summary

The successful publication of 80 excellent papers in 2023, the first year of the official conversion to an OA journal, would not have been possible without the support of the editorial board members, the editor-in-chief, and the contributions of authors and reviewers. Although the impact factor has dropped a bit from the previous year, it is believed that it will gradually increase.

4.2. Perspective

At present, there are some problems that we need to improve in the next step: the manuscript processing cycle is longer than other OA journals; the editorial board needs to be further expanded, and the promotion of the journal needs to be further improved.

Next year, everything will be better.

References

[1]	Effects of Rayleigh waves on the essential spectrum in perturbed doubly periodic elliptic problems. Integral Equations Operator Theory (2017) 88: 373-386.
[2]	F. L. Bakharev and J. Taskinen, Bands in the spectrum of a periodic elastic waveguide, Z. Angew. Math. Phys., 68 (2017), 27 pp. doi: 10.1007/s00033-017-0846-0
[3]	M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-adjoint Operators in Hilbert Space, D. Reidel Publishing Co., Dordrecht, 1987.
[4]	Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer. Asymptot. Anal. (2006) 50: 121-173.
[5]	Expansion in characteristic functions of an equation with periodic coefficients. Dokl. Akad. Nauk SSSR (1950) 73: 1117-1120.
[6]	Spectral properties of periodic media in the large coupling limit. Comm. Partial Differential Equations (2000) 25: 1445-1470.
[7]	R. Hempel and O. Post, Spectral gaps for periodic elliptic operators with high contrast: An overview, Progress in analysis, Vol. I, II (Berlin, 2001), 577–587.
[8]	Boundary-value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč. (1967) 16: 209-292.
[9]	Spectral properties of the operator bundles generated by elliptic boundary-value problems in a cone. translation in Funct. Anal. Appl. (1988) 22: 114-121.
[10]	V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Elliptic Boundary Value Problems in Domains With Point Singularities, Amer. Math. Soc., Providence RI, 1997.
[11]	Floquet theory for partial differential equations. Uspekhi Mat. Nauk (1982) 37: 3-52.
[12]	P. A. Kuchment, Floquet Theory for Partial Differential Equations, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8573-7
[13]	Artificial boundary conditions on polyhedral truncation surfaces for three-dimensional elasticity systems. Comptes Rendus Mécanique (2004) 332: 591-596.
[14]	J.–L. Lions and E. Magenes, Non-Homogeneus Boundary Value Problems and Applications, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.
[15]	On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points. Math. Nachr. (1977) 76: 29-60.
[16]	Estimates in $L_p$ 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. (1978) 81: 25-82.
[17]	The polynomial property of self-adjoint elliptic boundary-value problems and the algebraic description of their attributes. translation in Russian Math. Surveys (1999) 54: 947-1014.
[18]	Asymptotic behavior of the solution and the modeling of the Dirichlet problem in an angular domain with rapidly oscillating boundary. translation in St. Petersburg Math. J. (2008) 19: 297-326.
[19]	S. A. Nazarov, The Neumann problem in angular domains with periodic boundaries and parabolic perturbations of the boundaries, Tr. Mosk. Mat. Obs., 69(2008), 182–241; translation in Trans. Moscow Math. Soc., (2008), 153–208. doi: 10.1090/S0077-1554-08-00173-8
[20]	Gap in the essential spectrum of an elliptic formally self-adjoint system of differential equation. translation in Differ. Equ. (2010) 46: 730-741.
[21]	S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110848915.525
[22]	Essential spectrum of a periodic elastic waveguide may contain arbitrarily many gaps. Appl. Anal. (2010) 89: 109-124.
[23]	Spectral gaps for periodic piezoelectric waveguides. Z. Angew. Math. Phys. (2015) 66: 3017-3047.
[24]	J. Nečas, Les Méthodes in Théorie Des Équations Elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967.
[25]	M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators, Trudy Mat. Inst. Steklov., 171 (1985), 122 pp.
[26]	Eigenfunctions of the 2-dimensional anisotropic elasticity operator and algebraic equivalent materials. ZAMM Z. Angew. Math. Mech. (2008) 88: 100-115.
[27]	M. I. Vishik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with a small parameter, Uspehi Mat. Nauk (N.S.), 12 1957, 3–122.
[28]	On gaps in the spectrum of some elliptic operators in divergent form with periodic coefficients. translation in St. Petersburg Math. J. (2005) 16: 773-790.

This article has been cited by:

1.	Benedetto Piccoli, Nastassia Pouradier Duteil, 2021, Chapter 12, 978-3-030-82945-2, 289, 10.1007/978-3-030-82946-9_12
2.	Yurii Averboukh, Nonlocal Balance Equation: Representation and Approximation of Solution, 2024, 1040-7294, 10.1007/s10884-024-10373-8
3.	Rico Berner, Thilo Gross, Christian Kuehn, Jürgen Kurths, Serhiy Yanchuk, Adaptive dynamical networks, 2023, 1031, 03701573, 1, 10.1016/j.physrep.2023.08.001
4.	Marvin Lücke, Jobst Heitzig, Péter Koltai, Nora Molkenthin, Stefanie Winkelmann, Large population limits of Markov processes on random networks, 2023, 166, 03044149, 104220, 10.1016/j.spa.2023.09.007
5.	Immanuel Ben-Porat, José A. Carrillo, Pierre-Emmanuel Jabin, The graph limit for a pairwise competition model, 2024, 413, 00220396, 329, 10.1016/j.jde.2024.08.069
6.	Martin Gugat, Michael Herty, Chiara Segala, The turnpike property for mean-field optimal control problems, 2024, 35, 0956-7925, 733, 10.1017/S0956792524000044
7.	Immanuel Ben-Porat, José A. Carrillo, Sondre T. Galtung, Mean field limit for one dimensional opinion dynamics with Coulomb interaction and time dependent weights, 2024, 240, 0362546X, 113462, 10.1016/j.na.2023.113462
8.	Nathalie Ayi, Nastassia Pouradier Duteil, 2024, Chapter 3, 978-3-031-73422-9, 79, 10.1007/978-3-031-73423-6_3
9.	Antonio Esposito, Georg Heinze, André Schlichting, Graph-to-local limit for the nonlocal interaction equation, 2025, 00217824, 103663, 10.1016/j.matpur.2025.103663
10.	N. I. Pogodaev, M. V. Staritsyn, Optimal Control of Nonlocal Balance Equations in the Space of Nonnegative Measures, 2025, 66, 0037-4466, 576, 10.1134/S0037446625020223

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

Networks and Heterogeneous Media

1.2 1.8

Metrics

Article views(1845) PDF downloads(323) Cited by(1)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(3)

Networks and Heterogeneous Media

The band-gap structure of the spectrum in a periodic medium of masonry type

Related Papers:

Abstract

1. The Journal

2. Manuscript processing

2.1. Manuscript processing time

2.2. Distribution of countries

3. Journal development

3.1. Editorial board countries and regions

3.2. Article metrics

3.3. Special issues

4. Summary and perspective

4.1. Summary

4.2. Perspective

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

Networks and Heterogeneous Media

The band-gap structure of the spectrum in a periodic medium of masonry type

Related Papers:

Abstract

1. The Journal

2. Manuscript processing

2.1. Manuscript processing time

2.2. Distribution of countries

3. Journal development

3.1. Editorial board countries and regions

3.2. Article metrics

3.3. Special issues

4. Summary and perspective

4.1. Summary

4.2. Perspective

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog