American Institute of Mathematical Sciences

• Previous Article
Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems
• NHM Home
• This Issue
• Next Article
Preprocessing and analyzing genetic data with complex networks: An application to Obstructive Nephropathy
September  2012, 7(3): 483-501. doi: 10.3934/nhm.2012.7.483

Dirichlet to Neumann maps for infinite quantum graphs

Received  September 2011 Revised  June 2012 Published  October 2012

The Dirichlet problem and Dirichlet to Neumann map are analyzed for elliptic equations on a large collection of infinite quantum graphs. For a dense set of continuous functions on the graph boundary, the Dirichlet to Neumann map has values in the Radon measures on the graph boundary.
Citation: Robert Carlson. Dirichlet to Neumann maps for infinite quantum graphs. Networks and Heterogeneous Media, 2012, 7 (3) : 483-501. doi: 10.3934/nhm.2012.7.483
References:
 [1] G. Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numerical Functional Analysis and Optimization, 25 (2004), 321-348. doi: 10.1081/NFA-120039655. [2] S. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Probl. Imaging, 2 (2008), 1-21. doi: 10.3934/ipi.2008.2.1. [3] M. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems, 20 (2004), 647-672. doi: 10.1088/0266-5611/20/3/002. [4] M. Brown and R. Weikard, A Borg-Levinson theorem for trees, Proc. R. Soc. London Ser. A, 461 (2005), 3231-3243. doi: 10.1098/rspa.2005.1513. [5] A. Calderon, On an inverse boundary value problem, Computational and Applied Mathematics, 25 (2006), 133-138. doi: 10.1590/S0101-82052006000200002. [6] R. Carlson, Linear network models related to blood flow, in "Quantum Graphs and their Applications," Contemp. Math, 415 (2006), 65-80. doi: 10.1090/conm/415/07860. [7] R. Carlson, Boundary value problems for infinite metric graphs, in Analysis on Graphs and Its Applications, PSPM, 77 (2008), 355-368. [8] R. Carlson, After the explosion: Dirichlet forms and boundary problems for infinite graphs,, preprint, (). [9] E. Curtis, D. Ingerman and J. Morrow, Circular planar graphs and resistor networks, Linear Algebra Appl., 283 (1998), 115-150. doi: 10.1016/S0024-3795(98)10087-3. [10] P. Cartier, Fonctions harmoniques sur un arbre, Sympos. Math, 9 (1972), 203-270. [11] F. Chung, "Spectral Graph Theory,'' American Mathematical Society, Providence, 1997. [12] J. Cohen, F. Colonna and D. Singman, Distributions and measures on the boundary of a tree, Journal of Mathematical Analysis and Applications, 293 (2004), 89-107. doi: 10.1016/j.jmaa.2003.12.015. [13] Y. Colin de Verdiere, "Spectres de Graphes,'' Societe Mathematique de France, 1998. [14] Y. Colin de Verdiere, N. Torki-Hamza and F. Truc, Essential self-adjointness for combinatorial Schrödinger operators II-metrically noncomplete graphs, Mathematical Physics, Analysis, and Geometry, 14 (2011), 21-38. [15] P. Doyle and J. L. Snell, "Random Walks and Electric Networks,'' MAA, Washington, D. C., 1984. [16] P. Exner, J. Keating, P. Kuchment, T. Sunada and A. Teplaev, "Analysis on Graphs and Its Applications,'' American Mathematical Society, 2008. [17] G. Folland, "Real Analysis,'' John Wiley and Sons, New York, 1984. [18] A. Georgakopoulos, Graph topologies induced by edge lengths, Discrete Mathematics, 311 (2011), 1523-1542. doi: 10.1016/j.disc.2011.02.012. [19] J. Hocking and G. Young, "Topology,'' Addison-Wesley, 1961. [20] P. E. T. Jorgensen and E. P. J. Pearse, Operator theory and analysis of infinite networks,, preprint, (). [21] T. Kato, "Perturbation Theory for Linear Operators,'' Springer-Verlag, New York, 1995. [22] M. Keller and D. Lenz, Unbounded laplacians on graphs: Basic spectral properties and the heat equation, Math. Model. Nat. Phenom., 5 (2010), 198-224. doi: 10.1051/mmnp/20105409. [23] P. Lax, "Functional Analysis,'' Wiley, 2002. [24] R. Lyons and Y. Peres, "Probability on Trees and Networks,'', Cambridge University Press. In preparation. , (). [25] B. Maury, D. Salort and C. Vannier, Trace theorem for trees and application to the human lungs, Networks and Heterogeneous Media, 4 (2009), 469-500. [26] S. Nicaise, Some results on spectral theory over networks, applied to nerve impulse transmission, Springer Lecture Notes in Mathematics, 1171 (1985), 532-541. doi: 10.1007/BFb0076584. [27] H. Royden, "Real Analysis,'' Macmillan, New York, 1988. [28] J. Sylvester and G. Uhlmann, The Dirichlet to Neumann map and applications, Inverse problems in partial differential equations (Arcata, CA, 1989). SIAM, Philadelphia, 1990. [29] W. Woess, "Denumerable Markov Chains,'' European Mathematical Society, 2009. doi: 10.4171/071. [30] M. Picardello and W. Woess, Martin boundaries of random walks: ends of trees and groups, Trans. American Math. Soc., 302 (1987), 185-205. doi: 10.1090/S0002-9947-1987-0887505-2. [31] D. Zelig, "Properties of Solutions of Partial Differential Equations Defined on Human Lung Shaped Domains,'' Ph.D. Thesis, Department of Applied Mathematics, Technion - Israel Institute of Technology, 2005.

show all references

References:
 [1] G. Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numerical Functional Analysis and Optimization, 25 (2004), 321-348. doi: 10.1081/NFA-120039655. [2] S. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Probl. Imaging, 2 (2008), 1-21. doi: 10.3934/ipi.2008.2.1. [3] M. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems, 20 (2004), 647-672. doi: 10.1088/0266-5611/20/3/002. [4] M. Brown and R. Weikard, A Borg-Levinson theorem for trees, Proc. R. Soc. London Ser. A, 461 (2005), 3231-3243. doi: 10.1098/rspa.2005.1513. [5] A. Calderon, On an inverse boundary value problem, Computational and Applied Mathematics, 25 (2006), 133-138. doi: 10.1590/S0101-82052006000200002. [6] R. Carlson, Linear network models related to blood flow, in "Quantum Graphs and their Applications," Contemp. Math, 415 (2006), 65-80. doi: 10.1090/conm/415/07860. [7] R. Carlson, Boundary value problems for infinite metric graphs, in Analysis on Graphs and Its Applications, PSPM, 77 (2008), 355-368. [8] R. Carlson, After the explosion: Dirichlet forms and boundary problems for infinite graphs,, preprint, (). [9] E. Curtis, D. Ingerman and J. Morrow, Circular planar graphs and resistor networks, Linear Algebra Appl., 283 (1998), 115-150. doi: 10.1016/S0024-3795(98)10087-3. [10] P. Cartier, Fonctions harmoniques sur un arbre, Sympos. Math, 9 (1972), 203-270. [11] F. Chung, "Spectral Graph Theory,'' American Mathematical Society, Providence, 1997. [12] J. Cohen, F. Colonna and D. Singman, Distributions and measures on the boundary of a tree, Journal of Mathematical Analysis and Applications, 293 (2004), 89-107. doi: 10.1016/j.jmaa.2003.12.015. [13] Y. Colin de Verdiere, "Spectres de Graphes,'' Societe Mathematique de France, 1998. [14] Y. Colin de Verdiere, N. Torki-Hamza and F. Truc, Essential self-adjointness for combinatorial Schrödinger operators II-metrically noncomplete graphs, Mathematical Physics, Analysis, and Geometry, 14 (2011), 21-38. [15] P. Doyle and J. L. Snell, "Random Walks and Electric Networks,'' MAA, Washington, D. C., 1984. [16] P. Exner, J. Keating, P. Kuchment, T. Sunada and A. Teplaev, "Analysis on Graphs and Its Applications,'' American Mathematical Society, 2008. [17] G. Folland, "Real Analysis,'' John Wiley and Sons, New York, 1984. [18] A. Georgakopoulos, Graph topologies induced by edge lengths, Discrete Mathematics, 311 (2011), 1523-1542. doi: 10.1016/j.disc.2011.02.012. [19] J. Hocking and G. Young, "Topology,'' Addison-Wesley, 1961. [20] P. E. T. Jorgensen and E. P. J. Pearse, Operator theory and analysis of infinite networks,, preprint, (). [21] T. Kato, "Perturbation Theory for Linear Operators,'' Springer-Verlag, New York, 1995. [22] M. Keller and D. Lenz, Unbounded laplacians on graphs: Basic spectral properties and the heat equation, Math. Model. Nat. Phenom., 5 (2010), 198-224. doi: 10.1051/mmnp/20105409. [23] P. Lax, "Functional Analysis,'' Wiley, 2002. [24] R. Lyons and Y. Peres, "Probability on Trees and Networks,'', Cambridge University Press. In preparation. , (). [25] B. Maury, D. Salort and C. Vannier, Trace theorem for trees and application to the human lungs, Networks and Heterogeneous Media, 4 (2009), 469-500. [26] S. Nicaise, Some results on spectral theory over networks, applied to nerve impulse transmission, Springer Lecture Notes in Mathematics, 1171 (1985), 532-541. doi: 10.1007/BFb0076584. [27] H. Royden, "Real Analysis,'' Macmillan, New York, 1988. [28] J. Sylvester and G. Uhlmann, The Dirichlet to Neumann map and applications, Inverse problems in partial differential equations (Arcata, CA, 1989). SIAM, Philadelphia, 1990. [29] W. Woess, "Denumerable Markov Chains,'' European Mathematical Society, 2009. doi: 10.4171/071. [30] M. Picardello and W. Woess, Martin boundaries of random walks: ends of trees and groups, Trans. American Math. Soc., 302 (1987), 185-205. doi: 10.1090/S0002-9947-1987-0887505-2. [31] D. Zelig, "Properties of Solutions of Partial Differential Equations Defined on Human Lung Shaped Domains,'' Ph.D. Thesis, Department of Applied Mathematics, Technion - Israel Institute of Technology, 2005.
 [1] Ihsane Bikri, Ronald B. Guenther, Enrique A. Thomann. The Dirichlet to Neumann map - An application to the Stokes problem in half space. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 221-230. doi: 10.3934/dcdss.2010.3.221 [2] Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez. Parabolic problems with varying operators and Dirichlet and Neumann boundary conditions on varying sets. Conference Publications, 2007, 2007 (Special) : 181-190. doi: 10.3934/proc.2007.2007.181 [3] Roberto De Leo, James A. Yorke. The graph of the logistic map is a tower. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5243-5269. doi: 10.3934/dcds.2021075 [4] Shouchuan Hu, Nikolaos S. Papageorgiou. Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2889-2922. doi: 10.3934/cpaa.2013.12.2889 [5] Victor Isakov. Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 631-640. doi: 10.3934/dcdss.2011.4.631 [6] Jussi Behrndt, A. F. M. ter Elst. The Dirichlet-to-Neumann map for Schrödinger operators with complex potentials. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 661-671. doi: 10.3934/dcdss.2017033 [7] Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems and Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745 [8] Sunghan Kim, Ki-Ahm Lee, Henrik Shahgholian. Homogenization of the boundary value for the Dirichlet problem. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6843-6864. doi: 10.3934/dcds.2019234 [9] N. Arada, J.-P. Raymond. Time optimal problems with Dirichlet boundary controls. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1549-1570. doi: 10.3934/dcds.2003.9.1549 [10] Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems and Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959 [11] Victor Isakov, Jenn-Nan Wang. Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map. Inverse Problems and Imaging, 2014, 8 (4) : 1139-1150. doi: 10.3934/ipi.2014.8.1139 [12] 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 and Imaging, 2019, 13 (6) : 1213-1258. doi: 10.3934/ipi.2019054 [13] Mourad Bellassoued, Zouhour Rezig. Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1061-1084. doi: 10.3934/dcdss.2021158 [14] Gökhan Mutlu. On the quotient quantum graph with respect to the regular representation. Communications on Pure and Applied Analysis, 2021, 20 (2) : 885-902. doi: 10.3934/cpaa.2020295 [15] Liu Hui, Lin Zhi, Waqas Ahmad. Network(graph) data research in the coordinate system. Mathematical Foundations of Computing, 2018, 1 (1) : 1-10. doi: 10.3934/mfc.2018001 [16] Deena Schmidt, Janet Best, Mark S. Blumberg. Random graph and stochastic process contributions to network dynamics. Conference Publications, 2011, 2011 (Special) : 1279-1288. doi: 10.3934/proc.2011.2011.1279 [17] Vladimir P. Burskii, Alexei S. Zhedanov. On Dirichlet, Poncelet and Abel problems. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1587-1633. doi: 10.3934/cpaa.2013.12.1587 [18] Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569 [19] Dorina Mitrea, Marius Mitrea, Sylvie Monniaux. The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1295-1333. doi: 10.3934/cpaa.2008.7.1295 [20] Agnid Banerjee, Nicola Garofalo. On the Dirichlet boundary value problem for the normalized $p$-laplacian evolution. Communications on Pure and Applied Analysis, 2015, 14 (1) : 1-21. doi: 10.3934/cpaa.2015.14.1

2020 Impact Factor: 1.213