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September  2012, 7(3): 483-501. doi: 10.3934/nhm.2012.7.483

Dirichlet to Neumann maps for infinite quantum graphs

Robert Carlson 1,

1. 

Department of Mathematics, University of Colorado at Colorado Springs, Colorado Springs, CO 80933

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.
Keywords: infinite graph, Dirichlet problem, quantum graph., network boundary problems, Dirichlet to Neumann map.
Mathematics Subject Classification: Primary: 34B4.
Citation: Robert Carlson. Dirichlet to Neumann maps for infinite quantum graphs. Networks & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Calderon, On an inverse boundary value problem, Computational and Applied Mathematics, 25 (2006), 133-138. doi: 10.1590/S0101-82052006000200002.  Google Scholar

[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.  Google Scholar

[7]

R. Carlson, Boundary value problems for infinite metric graphs, in Analysis on Graphs and Its Applications, PSPM, 77 (2008), 355-368.  Google Scholar

[8]

R. Carlson, After the explosion: Dirichlet forms and boundary problems for infinite graphs,, preprint, ().   Google Scholar

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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.  Google Scholar

[10]

P. Cartier, Fonctions harmoniques sur un arbre, Sympos. Math, 9 (1972), 203-270.  Google Scholar

[11]

F. Chung, "Spectral Graph Theory,'' American Mathematical Society, Providence, 1997.  Google Scholar

[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.  Google Scholar

[13]

Y. Colin de Verdiere, "Spectres de Graphes,'' Societe Mathematique de France, 1998.  Google Scholar

[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.  Google Scholar

[15]

P. Doyle and J. L. Snell, "Random Walks and Electric Networks,'' MAA, Washington, D. C., 1984.  Google Scholar

[16]

P. Exner, J. Keating, P. Kuchment, T. Sunada and A. Teplaev, "Analysis on Graphs and Its Applications,'' American Mathematical Society, 2008.  Google Scholar

[17]

G. Folland, "Real Analysis,'' John Wiley and Sons, New York, 1984.  Google Scholar

[18]

A. Georgakopoulos, Graph topologies induced by edge lengths, Discrete Mathematics, 311 (2011), 1523-1542. doi: 10.1016/j.disc.2011.02.012.  Google Scholar

[19]

J. Hocking and G. Young, "Topology,'' Addison-Wesley, 1961.  Google Scholar

[20]

P. E. T. Jorgensen and E. P. J. Pearse, Operator theory and analysis of infinite networks,, preprint, ().   Google Scholar

[21]

T. Kato, "Perturbation Theory for Linear Operators,'' Springer-Verlag, New York, 1995.  Google Scholar

[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.  Google Scholar

[23]

P. Lax, "Functional Analysis,'' Wiley, 2002.  Google Scholar

[24]

R. Lyons and Y. Peres, "Probability on Trees and Networks,'', Cambridge University Press. In preparation. , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

H. Royden, "Real Analysis,'' Macmillan, New York, 1988.  Google Scholar

[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.  Google Scholar

[29]

W. Woess, "Denumerable Markov Chains,'' European Mathematical Society, 2009. doi: 10.4171/071.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Calderon, On an inverse boundary value problem, Computational and Applied Mathematics, 25 (2006), 133-138. doi: 10.1590/S0101-82052006000200002.  Google Scholar

[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.  Google Scholar

[7]

R. Carlson, Boundary value problems for infinite metric graphs, in Analysis on Graphs and Its Applications, PSPM, 77 (2008), 355-368.  Google Scholar

[8]

R. Carlson, After the explosion: Dirichlet forms and boundary problems for infinite graphs,, preprint, ().   Google Scholar

[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.  Google Scholar

[10]

P. Cartier, Fonctions harmoniques sur un arbre, Sympos. Math, 9 (1972), 203-270.  Google Scholar

[11]

F. Chung, "Spectral Graph Theory,'' American Mathematical Society, Providence, 1997.  Google Scholar

[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.  Google Scholar

[13]

Y. Colin de Verdiere, "Spectres de Graphes,'' Societe Mathematique de France, 1998.  Google Scholar

[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.  Google Scholar

[15]

P. Doyle and J. L. Snell, "Random Walks and Electric Networks,'' MAA, Washington, D. C., 1984.  Google Scholar

[16]

P. Exner, J. Keating, P. Kuchment, T. Sunada and A. Teplaev, "Analysis on Graphs and Its Applications,'' American Mathematical Society, 2008.  Google Scholar

[17]

G. Folland, "Real Analysis,'' John Wiley and Sons, New York, 1984.  Google Scholar

[18]

A. Georgakopoulos, Graph topologies induced by edge lengths, Discrete Mathematics, 311 (2011), 1523-1542. doi: 10.1016/j.disc.2011.02.012.  Google Scholar

[19]

J. Hocking and G. Young, "Topology,'' Addison-Wesley, 1961.  Google Scholar

[20]

P. E. T. Jorgensen and E. P. J. Pearse, Operator theory and analysis of infinite networks,, preprint, ().   Google Scholar

[21]

T. Kato, "Perturbation Theory for Linear Operators,'' Springer-Verlag, New York, 1995.  Google Scholar

[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.  Google Scholar

[23]

P. Lax, "Functional Analysis,'' Wiley, 2002.  Google Scholar

[24]

R. Lyons and Y. Peres, "Probability on Trees and Networks,'', Cambridge University Press. In preparation. , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

H. Royden, "Real Analysis,'' Macmillan, New York, 1988.  Google Scholar

[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.  Google Scholar

[29]

W. Woess, "Denumerable Markov Chains,'' European Mathematical Society, 2009. doi: 10.4171/071.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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