Dirichlet to Neumann maps for infinite quantum graphs

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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

Abstract
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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, ().
[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.

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