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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 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, arXiv:1109.3137.

[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, arXiv:0806.3881.

[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. http://mypage.iu.edu/~rdlyons

[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, arXiv:1109.3137.

[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, arXiv:0806.3881.

[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. http://mypage.iu.edu/~rdlyons

[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.

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