Embedded trace operator for infinite metric trees

We consider a class of infinite weighted metric trees obtained as perturbations of self-similar regular trees. Possible definitions of the boundary traces of functions in the Sobolev space on such a structure are discussed by using identifications of the tree boundary with a surface. Our approach unifies some constructions proposed by Maury, Salort, Vannier (2009) for dyadic discrete weighted trees (expansion in orthogonal bases of harmonic functions on the graph and using Haar-type bases on the domain representing the boundary), and by Nicaise, Semin (2018) and Joly, Kachanovska, Semin (2019) for fractal metric trees (approximation by finite sections and identification of the boundary with a interval): we show that both machineries give the same trace map, and for a range of parameters we establish the precise Sobolev regularity of the traces. In addition, we introduce new geometric ingredients by proposing an identification with arbitrary Riemannian manifolds. It is shown that any compact manifold admits a suitable multiscale decomposition and, therefore, can be identified with a metric tree boundary in the context of trace theorems.

[1]  Robert Carlson,et al.  Robin boundary conditions for the Laplacian on metric graph completions , 2021, 2112.04578.

[2]  Nageswari Shanmugalingam,et al.  Extension and trace results for doubling metric measure spaces and their hyperbolic fillings , 2020, Journal de Mathématiques Pures et Appliquées.

[3]  P. Joly,et al.  Local Transparent Boundary Conditions for Wave Propagation in Fractal Trees (II). Error and Complexity Analysis , 2020, SIAM J. Numer. Anal..

[4]  P. Koskela,et al.  Trace Operators on Regular Trees , 2020 .

[5]  P. Koskela,et al.  Trace and Density Results on Regular Trees , 2019, Potential Analysis.

[6]  Delio Mugnolo,et al.  Self‐adjoint and Markovian extensions of infinite quantum graphs , 2019, Journal of the London Mathematical Society.

[7]  Netanel Levi,et al.  On the Decomposition of the Laplacian on Metric Graphs , 2019, Annales Henri Poincaré.

[8]  H. Neidhardt,et al.  Spectral Theory of Infinite Quantum Graphs , 2018, Annales Henri Poincaré.

[9]  D. Rosca,et al.  Uniform refinable 3D grids of regular convex polyhedrons and balls , 2018, Acta Mathematica Hungarica.

[10]  S. Nicaise,et al.  Density and trace results in generalized fractal networks , 2018 .

[11]  Ting-Kam Leonard Wong,et al.  Random Walks and Induced Dirichlet Forms on Compact Spaces of Homogeneous Type , 2018, Analysis, Probability and Mathematical Physics on Fractals.

[12]  M. Waurick,et al.  Boundary systems and (skew‐)self‐adjoint operators on infinite metric graphs , 2013, 1308.2635.

[13]  D. Lenz,et al.  New Relations Between Discrete and Continuous Transition Operators on (Metric) Graphs , 2013, Integral Equations and Operator Theory.

[14]  James T. Gill,et al.  Geometric analysis on Cantor sets and trees , 2013, 1304.0566.

[15]  C. Schneider,et al.  Sobolev spaces on Riemannian manifolds with bounded geometry: General coordinates and traces , 2013, 1301.2539.

[16]  Konstantin Pankrashkin.  An example of unitary equivalence between self-adjoint extensions and their parameters , 2012, 1212.6798.

[17]  O. Post.  Boundary pairs associated with quadratic forms , 2012, 1210.4707.

[18]  D. Lenz,et al.  Unbounded quantum graphs with unbounded boundary conditions , 2012, 1205.1944.

[19]  O. Post.  Spectral Analysis on Graph-like Spaces , 2012 .

[20]  P. Joly,et al.  Mathematical and numerical modeling of wave propagation in fractal trees , 2011 .

[21]  Konstantin Pankrashkin.  Unitary dimension reduction for a class of self-adjoint extensions with applications to graph-like structures , 2011, Journal of Mathematical Analysis and Applications.

[22]  Daniel Lenz,et al.  Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions , 2011, 1103.3695.

[23]  Bertrand Maury,et al.  A 2-adic approach of the human respiratory tree , 2010, Networks Heterog. Media.

[24]  G. Wolansky,et al.  Spectral properties of Schrödinger operators on radial N-dimensional infinite trees , 2008 .

[25]  L. Tartar.  An Introduction to Sobolev Spaces and Interpolation Spaces , 2007 .

[26]  V. Geyler,et al.  SPECTRA OF SELF-ADJOINT EXTENSIONS AND APPLICATIONS TO SOLVABLE SCHRÖDINGER OPERATORS , 2006, math-ph/0611088.

[27]  Konstantin Pankrashkin.  Spectra of Schrödinger Operators on Equilateral Quantum Graphs , 2005, math-ph/0512090.

[28]  P. Exner.  The von Neumann way to treat systems of mixed dimensionality , 2005 .

[29]  J. Below,et al.  Harmonic functions on locally finite networks , 2004 .

[30]  V. Geyler,et al.  Scattering on compact manifolds with infinitely thin horns , 2002, math-ph/0205030.

[31]  W. Woess.  Random walks on infinite graphs and groups, by Wolfgang Woess, Cambridge Tracts , 2001 .

[32]  C. Cattaneo.  The spectrum of the continuous Laplacian on a graph , 1997 .

[33]  I. Fonseca,et al.  Equilibrium configurations of defective crystals , 1992 .

[34]  J. Below.  A characteristic equation associated to an eigenvalue problem on c2-networks , 1985 .

[35]  J. Moser.  On the volume elements on a manifold , 1965 .

[36]  J. Whitehead.  On C 1 -Complexes , 1940 .

[37]  Radoslaw K. Wojciechowski,et al.  Graphs and Discrete Dirichlet Spaces , 2021, Grundlehren der mathematischen Wissenschaften.

[38]  P. Joly,et al.  TRANSPARENT BOUNDARY CONDITIONS FOR THE WAVE PROPAGATION IN FRACTAL TREES , 2018 .

[39]  M. Mitrea,et al.  Multi-Layer Potentials and Boundary Problems , 2013 .

[40]  Wolfgang Woess,et al.  Random walks, boundaries and spectra , 2011 .

[41]  J. Griepentrog,et al.  A bi-Lipschitz continuous, volume preserving map from the unit ball onto a cube , 2008 .

[42]  A. Cohen.  Numerical Analysis of Wavelet Methods , 2003 .