Density and trace results in generalized fractal networks
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 1023-1049.

The first aim of this paper is to give different necessary and sufficient conditions that guarantee the density of the set of compactly supported functions into the Sobolev space of order one in infinite p-adic weighted trees. The second goal is to define properly a trace operator in this Sobolev space if it makes sense.

DOI : 10.1051/m2an/2018021
Classification : 35B53, 35B40, 35B65, 35J05
Mots-clés : Laplace equation, fractal, graph domain, Liouville property, boundary operator
Nicaise, Serge 1 ; Semin, Adrien 1

1 
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Nicaise, Serge; Semin, Adrien. Density and trace results in generalized fractal networks. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 1023-1049. doi : 10.1051/m2an/2018021. http://archive.numdam.org/articles/10.1051/m2an/2018021/

[1] Y. Achdou and N. Tchou, Trace results on domains with self-similar fractal boundaries. J. Math. Pures Appl. 89 (2008) 596–623. | DOI | MR | Zbl

[2] Y. Achdou and N. Tchou, Trace theorems for a class of ramified domains with self-similar fractal boundaries. SIAM J. Math. Anal. 42 (2010) 1449–1482. | DOI | MR | Zbl

[3] Y. Achdou and T. Deheuvels, A transmission problem across a fractal self-similar interface. Multiscale Model. Simul. 14 (2016) 708–736. | DOI | MR | Zbl

[4] Y. Achdou, C. Sabot and N. Tchou, Diffusion and propagation problems in some ramified domains with a fractal boundary. ESAIM: M2AN 40 (2006) 623–652. | DOI | Numdam | MR | Zbl

[5] Y. Achdou, F. Camilli, A. Cutrì and N. Tchou, Hamilton–jacobi equations constrained on networks. Nonlinear Differ. Equ. Appl. NoDEA 20 (2013) 413–445. | DOI | MR | Zbl

[6] C. Anné and N. Torki-Hamza. The Gauss-Bonnet operator of an infinite graph. Anal. Math. Phys. 5 (2015) 137–159. | DOI | MR | Zbl

[7] G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1 - D Hyperbolic Systems. PNLDE Subseries in Control. Birkhäuser, Basel (2016). | MR | Zbl

[8] F. Bernicot, B. Maury and D. Salort, A 2-adic approach of the human respiratory tree. Netw. Heterog. Media 5 (2010) 405–422. | DOI | MR | Zbl

[9] A. Bressan, S. Čanić, M. Garavello, M. Herty and B. Piccoli, Flows on networks: recent results and perspectives. EMS Surv. Math. Sci. 1 (2014) 47–111. | DOI | MR | Zbl

[10] R. Capitanelli and M.A. Vivaldi, Uniform weighted estimates on pre-fractal domains. Discret. Contin. Dyn. Syst. Ser. B 19 (2014) 1969–1985. | MR | Zbl

[11] J. Carmesin, A characterization of the locally finite networks admitting non-constant harmonic functions of finite energy. Potential Anal. 37 (2012) 229–245. | DOI | MR | Zbl

[12] C. D’Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains. A continuous approach. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2010). | MR | Zbl

[13] H. Flanders, Infinite networks. I: resistive networks. IEEE Trans. Circuit Theory CT-18 (1971) 26–331. | MR | Zbl

[14] M. Garavello and B. Piccoli, Conservation laws models, in Traffic flow on networks. Vol. 1 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2006). | MR | Zbl

[15] A. Georgakopoulos, Uniqueness of electrical currents in a network of finite total resistance. J. Lond. Math. Soc. 82 (2010) 256–272. | DOI | MR | Zbl

[16] P. Joly and A. Semin, Mathematical and numerical modeling of wave propagation in fractal trees. C.R. Math. 349 (2011) 1047–1051. | DOI | MR | Zbl

[17] P. Joly and A. Semin, Wave propagation in fractal trees. mathematical and numerical issues. Netw. Heterog. Media (Submitted). | MR

[18] R. Lyons and Y. Peres, Probability on Trees and Networks. Vol. 42 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, New York (2016). | MR | Zbl

[19] B. Maury, D. Salort and C. Vannier, Trace theorems for trees, application to the human lungs. Netw. Hetereg. Media 4 (2009) 469–500. | DOI | MR | Zbl

[20] Paolo M. Soardi, Potential theory on infinite networks. Vol. 1590 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1994). | MR | Zbl

[21] P.M. Soardi and W. Woess, Uniqueness of currents in infinite resistive networks. Discret. Appl. Math. 31 (1991)37–49. | DOI | MR | Zbl

[22] P.M. Soardi and M. Yamasaki, Classification of infinite networks and its applications. Circuits Syst. Signal Process 12 (1993) 133–149. | DOI | Zbl

[23] C. Thomassen, Resistances and currents in infinite electrical networks. J. Comb. Theory, Ser. B 49 (1990) 87–102. | DOI | MR | Zbl

[24] J. Von Below and J. Lubary, Harmonic functions on locally finite networks. Results Math., 45 (2004) 1–20. | DOI | MR | Zbl

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