# American Institute of Mathematical Sciences

• Previous Article
Computing the asymptotic spectrum for networks representing energy landscapes using the minimum spanning tree
• NHM Home
• This Issue
• Next Article
Optimization of bodies with locally periodic microstructure by varying the periodicity pattern
September  2014, 9(3): 417-432. doi: 10.3934/nhm.2014.9.417

## Finite mechanical proxies for a class of reducible continuum systems

 1 Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste, 63 - 35121 Padova, Italy, Italy

Received  June 2013 Revised  May 2014 Published  October 2014

We present the exact finite reduction of a class of nonlinearly perturbed wave equations --typically, a non-linear elastic string-- based on the Amann--Conley--Zehnder paradigm. By solving an inverse eigenvalue problem, we establish an equivalence between the spectral finite description derived from A--C--Z and a discrete mechanical model, a well definite finite spring--mass system. By doing so, we decrypt the abstract information encoded in the finite reduction and obtain a physically sound proxy for the continuous problem.
Citation: Franco Cardin, Alberto Lovison. Finite mechanical proxies for a class of reducible continuum systems. Networks & Heterogeneous Media, 2014, 9 (3) : 417-432. doi: 10.3934/nhm.2014.9.417
##### References:
 [1] H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7 (1980), 539. Google Scholar [2] H. Amann and E. Zehnder, Multiple periodic solutions for a class of nonlinear autonomous wave equations,, Houston J. Math., 7 (1981), 147. Google Scholar [3] H. Amann, Multiple positive fixed points of asymptotically linear maps,, J. Functional Analysis, 17 (1974), 174. doi: 10.1016/0022-1236(74)90011-1. Google Scholar [4] H. Amann, Saddle points and multiple solutions of differential equations,, Math. Z., 169 (1979), 127. doi: 10.1007/BF01215273. Google Scholar [5] A. Ambrosetti, Critical points and nonlinear variational problems,, Mém. Soc. Math. France (N.S.), (1992). Google Scholar [6] S. S. Antman, The equations for large vibrations of strings,, Amer. Math. Monthly, 87 (1980), 359. doi: 10.2307/2321203. Google Scholar [7] A. Arsie and C. Ebenbauer, Locating omega-limit sets using height functions,, Journal of Differential Equations, 248 (2010), 2458. doi: 10.1016/j.jde.2009.11.012. Google Scholar [8] D. Bambusi and S. Paleari, Families of periodic solutions of resonant PDEs,, J. Nonlinear Sci., 11 (2001), 69. doi: 10.1007/s003320010010. Google Scholar [9] J. Berkovits, H. Leinfelder and V. Mustonen, Existence and multiplicity results for wave equations with time-independent nonlinearity,, Topol. Methods Nonlinear Anal., 22 (2003), 273. Google Scholar [10] M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities,, Comm. Math. Phys., 243 (2003), 315. doi: 10.1007/s00220-003-0972-8. Google Scholar [11] C. Boldrighini, A. De Masi, A. Pellegrinotti and E. Presutti, Collective phenomena in interacting particle systems,, Stochastic Process. Appl., 25 (1987), 137. doi: 10.1016/0304-4149(87)90194-3. Google Scholar [12] H. Brézis, J.-M. Coron and L. Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz,, Comm. Pure Appl. Math., 33 (1980), 667. doi: 10.1002/cpa.3160330507. Google Scholar [13] M. Cappiello, Pseudodifferential parametrices of infinite order for SG-hyperbolic problems,, Rend. Sem. Mat. Univ. Politec. Torino, 61 (2003), 411. Google Scholar [14] F. Cardin, Global finite generating functions for field theory,, in Classical and quantum integrability (Warsaw, 59 (2001), 133. doi: 10.4064/bc59-0-6. Google Scholar [15] F. Cardin and A. Lovison, Microscopic structures from reduction of continuum nonlinear problems,, AAPP - Physical, 91 (2013), 1. doi: 10.1478/AAPP.91S1A4. Google Scholar [16] F. Cardin and C. Tebaldi, Finite reductions for dissipative systems and viscous fluid-dynamic models on $\mathbbT^2$,, J. Math. Anal. Appl., 345 (2008), 213. doi: 10.1016/j.jmaa.2008.04.012. Google Scholar [17] F. Cardin, A. Lovison and M. Putti, Implementation of an exact finite reduction scheme for steady-state reaction-diffusion equations,, Internat. J. Numer. Methods Engrg., 69 (2007), 1804. doi: 10.1002/nme.1824. Google Scholar [18] H. Chiba, Continuous limit and the moments system for the globally coupled phase oscillators,, Discrete and Continuous Dynamical Systems - Series A, 33 (2013), 1891. doi: 10.3934/dcds.2013.33.1891. Google Scholar [19] M. Cicalese, A. DeSimone and C. I. Zeppieri, Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers,, Networks and Heterogeneous Media, 4 (2009), 667. doi: 10.3934/nhm.2009.4.667. Google Scholar [20] C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol'd,, Invent. Math., 73 (1983), 33. doi: 10.1007/BF01393824. Google Scholar [21] C. Conley, Isolated Invariant Sets and The Morse Index,, Number 38 in Regional conferences series in mathematics. Conference Board for the Mathematical Sciences, (1976). Google Scholar [22] J.-M. Coron, Periodic solutions of a nonlinear wave equation without assumption of monotonicity,, Math. Ann., 262 (1983), 273. doi: 10.1007/BF01455317. Google Scholar [23] R. de la Llave, Variational methods for quasi-periodic solutions of partial differential equations,, in Hamiltonian systems and celestial mechanics (Pátzcuaro, 6 (1998), 214. doi: 10.1142/9789812792099_0013. Google Scholar [24] M. Degiovanni, On Morse theory for continuous functionals,, Conf. Semin. Mat. Univ. Bari, 290 (2003), 1. Google Scholar [25] A. Di Carlo, private, communication., (). Google Scholar [26] C. Ebenbauer and A. Arsie, On an eigenflow equation and its Lie algebraic generalization,, Communications in Information and Systems, 8 (2008), 147. doi: 10.4310/CIS.2008.v8.n2.a6. Google Scholar [27] J. Frenkel and T. Kontorova, On the theory of plastic deformation and twinning,, Acad. Sci. U.S.S.R. J. Phys., 1 (1939), 137. Google Scholar [28] G. M. L. Gladwell, Inverse Problems in Vibration,, 119 of Solid Mechanics and its Applications. Kluwer Academic Publishers, (2004). Google Scholar [29] J. M. Greenberg and A. Nachman, Continuum limits for discrete gases with long- and short-range interactions,, Communications on Pure and Applied Mathematics, 47 (1994), 1239. doi: 10.1002/cpa.3160470905. Google Scholar [30] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,, Philosophical Magazine Series 5, 39 (1895), 422. doi: 10.1080/14786449508620739. Google Scholar [31] A. Lovison, Generating functions and finite parameter reductions in fields theory,, Bollettino Della Unione Matematica Italiana, 8A (2005), 569. Google Scholar [32] A. Lovison, F. Cardin and A. Bobbo, Discrete structures equivalent to nonlinear Dirichlet and wave equations,, Continuum Mech Therm, 21 (2009), 27. doi: 10.1007/s00161-009-0097-1. Google Scholar [33] M. Lucia, P. Magrone and H.-S. Zhou, A Dirichlet problem with asymptotically linear and changing sign nonlinearity,, Rev. Mat. Complut., 16 (2003), 465. Google Scholar [34] L. Maragliano, A. Fischer, E. Vanden-Eijnden and G. Ciccotti, String method in collective variables: Minimum free energy paths and isocommittor surfaces,, Journal of Chemical Physics, 125 (2006). doi: 10.1063/1.2212942. Google Scholar [35] S. Müller and A. Schlömerkemper, Discrete-to-continuum limit of magnetic forces,, Comptes Rendus Mathematique, 335 (2002), 393. doi: 10.1016/S1631-073X(02)02494-9. Google Scholar [36] L. Nirenberg, Variational and topological methods in nonlinear problems,, Bull. Amer. Math. Soc. (N.S.), 4 (1981), 267. doi: 10.1090/S0273-0979-1981-14888-6. Google Scholar [37] P. Nylen and F. Uhlig, Inverse eigenvalue problem: Existence of special spring - mass systems,, Inverse Problems, 13 (1997), 1071. doi: 10.1088/0266-5611/13/4/012. Google Scholar [38] P. H. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations,, Comm. Pure Appl. Math., 20 (1967), 145. doi: 10.1002/cpa.3160200105. Google Scholar [39] P. H. Rabinowitz, Free vibrations for a semilinear wave equation,, Comm. Pure Appl. Math., 31 (1978), 31. doi: 10.1002/cpa.3160310103. Google Scholar [40] S. Rybicki, Periodic solutions of vibrating strings. Degree theory approach,, Ann. Mat. Pura Appl. (4), 179 (2001), 197. doi: 10.1007/BF02505955. Google Scholar [41] M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators,, Holden-Day Inc., (1964). Google Scholar [42] C. Viterbo, Recent progress in periodic orbits of autonomous Hamiltonian systems and applications to symplectic geometry,, in Nonlinear functional analysis (Newark, 121 (1987), 227. Google Scholar [43] V. Volterra, Leçons sur les Fonctions de Ligne,, Gauthier-Villars, (1913). Google Scholar [44] J. von Neumann, Proposal and Analysis of a New Numerical Method for the Treatment of Hydrodynamical Shock Problems,, AMP Report, (1944), 1. Google Scholar [45] I. R. Yukhnovskiĭ, Phase Transitions of the Second Order,, World Scientific Publishing Co., (1987). doi: 10.1142/0289. Google Scholar [46] N. J. Zabusky and M. D. Kruskal, Interaction of "solitons'' in a collisionless plasma and the recurrence of initial states,, Phys. Rev. Lett., 15 (1965), 240. doi: 10.1103/PhysRevLett.15.240. Google Scholar

show all references

##### References:
 [1] H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7 (1980), 539. Google Scholar [2] H. Amann and E. Zehnder, Multiple periodic solutions for a class of nonlinear autonomous wave equations,, Houston J. Math., 7 (1981), 147. Google Scholar [3] H. Amann, Multiple positive fixed points of asymptotically linear maps,, J. Functional Analysis, 17 (1974), 174. doi: 10.1016/0022-1236(74)90011-1. Google Scholar [4] H. Amann, Saddle points and multiple solutions of differential equations,, Math. Z., 169 (1979), 127. doi: 10.1007/BF01215273. Google Scholar [5] A. Ambrosetti, Critical points and nonlinear variational problems,, Mém. Soc. Math. France (N.S.), (1992). Google Scholar [6] S. S. Antman, The equations for large vibrations of strings,, Amer. Math. Monthly, 87 (1980), 359. doi: 10.2307/2321203. Google Scholar [7] A. Arsie and C. Ebenbauer, Locating omega-limit sets using height functions,, Journal of Differential Equations, 248 (2010), 2458. doi: 10.1016/j.jde.2009.11.012. Google Scholar [8] D. Bambusi and S. Paleari, Families of periodic solutions of resonant PDEs,, J. Nonlinear Sci., 11 (2001), 69. doi: 10.1007/s003320010010. Google Scholar [9] J. Berkovits, H. Leinfelder and V. Mustonen, Existence and multiplicity results for wave equations with time-independent nonlinearity,, Topol. Methods Nonlinear Anal., 22 (2003), 273. Google Scholar [10] M. Berti and P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities,, Comm. Math. Phys., 243 (2003), 315. doi: 10.1007/s00220-003-0972-8. Google Scholar [11] C. Boldrighini, A. De Masi, A. Pellegrinotti and E. Presutti, Collective phenomena in interacting particle systems,, Stochastic Process. Appl., 25 (1987), 137. doi: 10.1016/0304-4149(87)90194-3. Google Scholar [12] H. Brézis, J.-M. Coron and L. Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz,, Comm. Pure Appl. Math., 33 (1980), 667. doi: 10.1002/cpa.3160330507. Google Scholar [13] M. Cappiello, Pseudodifferential parametrices of infinite order for SG-hyperbolic problems,, Rend. Sem. Mat. Univ. Politec. Torino, 61 (2003), 411. Google Scholar [14] F. Cardin, Global finite generating functions for field theory,, in Classical and quantum integrability (Warsaw, 59 (2001), 133. doi: 10.4064/bc59-0-6. Google Scholar [15] F. Cardin and A. Lovison, Microscopic structures from reduction of continuum nonlinear problems,, AAPP - Physical, 91 (2013), 1. doi: 10.1478/AAPP.91S1A4. Google Scholar [16] F. Cardin and C. Tebaldi, Finite reductions for dissipative systems and viscous fluid-dynamic models on $\mathbbT^2$,, J. Math. Anal. Appl., 345 (2008), 213. doi: 10.1016/j.jmaa.2008.04.012. Google Scholar [17] F. Cardin, A. Lovison and M. Putti, Implementation of an exact finite reduction scheme for steady-state reaction-diffusion equations,, Internat. J. Numer. Methods Engrg., 69 (2007), 1804. doi: 10.1002/nme.1824. Google Scholar [18] H. Chiba, Continuous limit and the moments system for the globally coupled phase oscillators,, Discrete and Continuous Dynamical Systems - Series A, 33 (2013), 1891. doi: 10.3934/dcds.2013.33.1891. Google Scholar [19] M. Cicalese, A. DeSimone and C. I. Zeppieri, Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers,, Networks and Heterogeneous Media, 4 (2009), 667. doi: 10.3934/nhm.2009.4.667. Google Scholar [20] C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol'd,, Invent. Math., 73 (1983), 33. doi: 10.1007/BF01393824. Google Scholar [21] C. Conley, Isolated Invariant Sets and The Morse Index,, Number 38 in Regional conferences series in mathematics. Conference Board for the Mathematical Sciences, (1976). Google Scholar [22] J.-M. Coron, Periodic solutions of a nonlinear wave equation without assumption of monotonicity,, Math. Ann., 262 (1983), 273. doi: 10.1007/BF01455317. Google Scholar [23] R. de la Llave, Variational methods for quasi-periodic solutions of partial differential equations,, in Hamiltonian systems and celestial mechanics (Pátzcuaro, 6 (1998), 214. doi: 10.1142/9789812792099_0013. Google Scholar [24] M. Degiovanni, On Morse theory for continuous functionals,, Conf. Semin. Mat. Univ. Bari, 290 (2003), 1. Google Scholar [25] A. Di Carlo, private, communication., (). Google Scholar [26] C. Ebenbauer and A. Arsie, On an eigenflow equation and its Lie algebraic generalization,, Communications in Information and Systems, 8 (2008), 147. doi: 10.4310/CIS.2008.v8.n2.a6. Google Scholar [27] J. Frenkel and T. Kontorova, On the theory of plastic deformation and twinning,, Acad. Sci. U.S.S.R. J. Phys., 1 (1939), 137. Google Scholar [28] G. M. L. Gladwell, Inverse Problems in Vibration,, 119 of Solid Mechanics and its Applications. Kluwer Academic Publishers, (2004). Google Scholar [29] J. M. Greenberg and A. Nachman, Continuum limits for discrete gases with long- and short-range interactions,, Communications on Pure and Applied Mathematics, 47 (1994), 1239. doi: 10.1002/cpa.3160470905. Google Scholar [30] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,, Philosophical Magazine Series 5, 39 (1895), 422. doi: 10.1080/14786449508620739. Google Scholar [31] A. Lovison, Generating functions and finite parameter reductions in fields theory,, Bollettino Della Unione Matematica Italiana, 8A (2005), 569. Google Scholar [32] A. Lovison, F. Cardin and A. Bobbo, Discrete structures equivalent to nonlinear Dirichlet and wave equations,, Continuum Mech Therm, 21 (2009), 27. doi: 10.1007/s00161-009-0097-1. Google Scholar [33] M. Lucia, P. Magrone and H.-S. Zhou, A Dirichlet problem with asymptotically linear and changing sign nonlinearity,, Rev. Mat. Complut., 16 (2003), 465. Google Scholar [34] L. Maragliano, A. Fischer, E. Vanden-Eijnden and G. Ciccotti, String method in collective variables: Minimum free energy paths and isocommittor surfaces,, Journal of Chemical Physics, 125 (2006). doi: 10.1063/1.2212942. Google Scholar [35] S. Müller and A. Schlömerkemper, Discrete-to-continuum limit of magnetic forces,, Comptes Rendus Mathematique, 335 (2002), 393. doi: 10.1016/S1631-073X(02)02494-9. Google Scholar [36] L. Nirenberg, Variational and topological methods in nonlinear problems,, Bull. Amer. Math. Soc. (N.S.), 4 (1981), 267. doi: 10.1090/S0273-0979-1981-14888-6. Google Scholar [37] P. Nylen and F. Uhlig, Inverse eigenvalue problem: Existence of special spring - mass systems,, Inverse Problems, 13 (1997), 1071. doi: 10.1088/0266-5611/13/4/012. Google Scholar [38] P. H. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations,, Comm. Pure Appl. Math., 20 (1967), 145. doi: 10.1002/cpa.3160200105. Google Scholar [39] P. H. Rabinowitz, Free vibrations for a semilinear wave equation,, Comm. Pure Appl. Math., 31 (1978), 31. doi: 10.1002/cpa.3160310103. Google Scholar [40] S. Rybicki, Periodic solutions of vibrating strings. Degree theory approach,, Ann. Mat. Pura Appl. (4), 179 (2001), 197. doi: 10.1007/BF02505955. Google Scholar [41] M. M. Vainberg, Variational Methods for the Study of Nonlinear Operators,, Holden-Day Inc., (1964). Google Scholar [42] C. Viterbo, Recent progress in periodic orbits of autonomous Hamiltonian systems and applications to symplectic geometry,, in Nonlinear functional analysis (Newark, 121 (1987), 227. Google Scholar [43] V. Volterra, Leçons sur les Fonctions de Ligne,, Gauthier-Villars, (1913). Google Scholar [44] J. von Neumann, Proposal and Analysis of a New Numerical Method for the Treatment of Hydrodynamical Shock Problems,, AMP Report, (1944), 1. Google Scholar [45] I. R. Yukhnovskiĭ, Phase Transitions of the Second Order,, World Scientific Publishing Co., (1987). doi: 10.1142/0289. Google Scholar [46] N. J. Zabusky and M. D. Kruskal, Interaction of "solitons'' in a collisionless plasma and the recurrence of initial states,, Phys. Rev. Lett., 15 (1965), 240. doi: 10.1103/PhysRevLett.15.240. Google Scholar
 [1] Anna Doubova, Enrique Fernández-Cara. Some geometric inverse problems for the linear wave equation. Inverse Problems & Imaging, 2015, 9 (2) : 371-393. doi: 10.3934/ipi.2015.9.371 [2] Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121 [3] Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems & Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019 [4] Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Perturbations of nonlinear eigenvalue problems. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1403-1431. doi: 10.3934/cpaa.2019068 [5] Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems & Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139 [6] Chuang Zheng. Inverse problems for the fourth order Schrödinger equation on a finite domain. Mathematical Control & Related Fields, 2015, 5 (1) : 177-189. doi: 10.3934/mcrf.2015.5.177 [7] Sari Lasanen. Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns. Inverse Problems & Imaging, 2012, 6 (2) : 267-287. doi: 10.3934/ipi.2012.6.267 [8] Sari Lasanen. Non-Gaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems & Imaging, 2012, 6 (2) : 215-266. doi: 10.3934/ipi.2012.6.215 [9] Li-Fang Dai, Mao-Lin Liang, Wei-Yuan Ma. Optimization problems on the rank of the solution to left and right inverse eigenvalue problem. Journal of Industrial & Management Optimization, 2015, 11 (1) : 171-183. doi: 10.3934/jimo.2015.11.171 [10] Leszek Gasiński, Nikolaos S. Papageorgiou. Multiple solutions for a class of nonlinear Neumann eigenvalue problems. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1491-1512. doi: 10.3934/cpaa.2014.13.1491 [11] Qun Lin, Hehu Xie. Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Problems & Imaging, 2013, 7 (3) : 795-811. doi: 10.3934/ipi.2013.7.795 [12] Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002 [13] Tetsutaro Shibata. Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2139-2147. doi: 10.3934/cpaa.2018102 [14] Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065 [15] Tai-Chia Lin. Vortices for the nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 391-398. doi: 10.3934/dcds.1999.5.391 [16] Jose Carlos Camacho, Maria de los Santos Bruzon. Similarity reductions of a nonlinear model for vibrations of beams. Conference Publications, 2015, 2015 (special) : 176-184. doi: 10.3934/proc.2015.0176 [17] Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831 [18] Jibin Li, Yi Zhang. Exact solitary wave and quasi-periodic wave solutions for four fifth-order nonlinear wave equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 623-631. doi: 10.3934/dcdsb.2010.13.623 [19] Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901 [20] Arnaud Heibig, Mohand Moussaoui. Exact controllability of the wave equation for domains with slits and for mixed boundary conditions. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 367-386. doi: 10.3934/dcds.1996.2.367

2018 Impact Factor: 0.871